%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP063^2 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n002.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:55:15 AM UTC 2026
% Result : Theorem 2.72s 1.66s
% Output : Refutation 2.72s
% Verified :
% SZS Type : Refutation
% Derivation depth : 9
% Number of leaves : 4
% Syntax : Number of formulae : 24 ( 16 unt; 0 typ; 0 def)
% Number of atoms : 89 ( 25 equ; 0 cnn)
% Maximal formula atoms : 3 ( 3 avg)
% Number of connectives : 376 ( 17 ~; 9 |; 3 &; 344 @)
% ( 0 <=>; 3 =>; 0 <=; 0 <~>)
% Maximal formula depth : 12 ( 6 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 70 ( 67 usr; 11 con; 0-4 aty)
% Number of variables : 18 ( 0 ^; 18 !; 0 ?; 18 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_5,type,
nat: $tType ).
thf(type_def_6,type,
real: $tType ).
thf(func_def_0,type,
real: $tType ).
thf(func_def_1,type,
nat: $tType ).
thf(func_def_2,type,
type:
!>[X0: $tType] : $o ).
thf(func_def_3,type,
one:
!>[X0: $tType] : $o ).
thf(func_def_4,type,
ring:
!>[X0: $tType] : $o ).
thf(func_def_5,type,
zero:
!>[X0: $tType] : $o ).
thf(func_def_6,type,
minus:
!>[X0: $tType] : $o ).
thf(func_def_7,type,
semiring:
!>[X0: $tType] : $o ).
thf(func_def_8,type,
neg_numeral:
!>[X0: $tType] : $o ).
thf(func_def_9,type,
mult_zero:
!>[X0: $tType] : $o ).
thf(func_def_10,type,
group_add:
!>[X0: $tType] : $o ).
thf(func_def_11,type,
monoid_add:
!>[X0: $tType] : $o ).
thf(func_def_12,type,
monoid_mult:
!>[X0: $tType] : $o ).
thf(func_def_13,type,
idom_abs_sgn:
!>[X0: $tType] : $o ).
thf(func_def_14,type,
zero_neq_one:
!>[X0: $tType] : $o ).
thf(func_def_15,type,
ab_group_add:
!>[X0: $tType] : $o ).
thf(func_def_16,type,
comm_semiring:
!>[X0: $tType] : $o ).
thf(func_def_17,type,
semigroup_add:
!>[X0: $tType] : $o ).
thf(func_def_18,type,
semigroup_mult:
!>[X0: $tType] : $o ).
thf(func_def_19,type,
linordered_idom:
!>[X0: $tType] : $o ).
thf(func_def_20,type,
comm_monoid_add:
!>[X0: $tType] : $o ).
thf(func_def_21,type,
ordered_semiring:
!>[X0: $tType] : $o ).
thf(func_def_22,type,
ab_semigroup_add:
!>[X0: $tType] : $o ).
thf(func_def_23,type,
comm_monoid_diff:
!>[X0: $tType] : $o ).
thf(func_def_24,type,
comm_monoid_mult:
!>[X0: $tType] : $o ).
thf(func_def_25,type,
ab_semigroup_mult:
!>[X0: $tType] : $o ).
thf(func_def_26,type,
linordered_semidom:
!>[X0: $tType] : $o ).
thf(func_def_27,type,
cancel_semigroup_add:
!>[X0: $tType] : $o ).
thf(func_def_28,type,
ordered_ab_group_add:
!>[X0: $tType] : $o ).
thf(func_def_29,type,
comm_s1003936772cancel:
!>[X0: $tType] : $o ).
thf(func_def_30,type,
cancel1352612707id_add:
!>[X0: $tType] : $o ).
thf(func_def_31,type,
ring_11004092258visors:
!>[X0: $tType] : $o ).
thf(func_def_32,type,
cancel146912293up_add:
!>[X0: $tType] : $o ).
thf(func_def_33,type,
linord219039673up_add:
!>[X0: $tType] : $o ).
thf(func_def_34,type,
ordere216010020id_add:
!>[X0: $tType] : $o ).
thf(func_def_35,type,
ordere142940540dd_abs:
!>[X0: $tType] : $o ).
thf(func_def_36,type,
ordere779506340up_add:
!>[X0: $tType] : $o ).
thf(func_def_37,type,
semiri1193490041visors:
!>[X0: $tType] : $o ).
thf(func_def_38,type,
linord1659791738miring:
!>[X0: $tType] : $o ).
thf(func_def_39,type,
canoni770627133id_add:
!>[X0: $tType] : $o ).
thf(func_def_40,type,
ordere236663937imp_le:
!>[X0: $tType] : $o ).
thf(func_def_41,type,
ordere623563068d_diff:
!>[X0: $tType] : $o ).
thf(func_def_42,type,
semiri1923998003cancel:
!>[X0: $tType] : $o ).
thf(func_def_43,type,
ordere516151231imp_le:
!>[X0: $tType] : $o ).
thf(func_def_44,type,
genClo1161277105lle_PC: nat > real > real ).
thf(func_def_45,type,
genClo1278781456e_beta: real ).
thf(func_def_46,type,
genClo872980712lle_mu: real ).
thf(func_def_47,type,
genClo1144207539le_rho: real ).
thf(func_def_48,type,
genClo1015804716orrect: nat > real > $o ).
thf(func_def_49,type,
genClo293725282kRead2: ( nat > real ) > ( nat > real ) > real > ( nat > $o ) > $o ).
thf(func_def_50,type,
genClo1650508560e_rmax: real ).
thf(func_def_51,type,
genClo1651033342e_rmin: real ).
thf(func_def_52,type,
genClo1163638703lle_te: nat > nat > real ).
thf(func_def_53,type,
abs_abs:
!>[X0: $tType] : ( X0 > X0 ) ).
thf(func_def_54,type,
minus_minus:
!>[X0: $tType] : ( X0 > X0 > X0 ) ).
thf(func_def_55,type,
one_one:
!>[X0: $tType] : X0 ).
thf(func_def_56,type,
plus_plus:
!>[X0: $tType] : ( X0 > X0 > X0 ) ).
thf(func_def_57,type,
times_times:
!>[X0: $tType] : ( X0 > X0 > X0 ) ).
thf(func_def_58,type,
zero_zero:
!>[X0: $tType] : X0 ).
thf(func_def_59,type,
ord_less_eq:
!>[X0: $tType] : ( X0 > X0 > $o ) ).
thf(func_def_60,type,
i: nat ).
thf(func_def_61,type,
l: nat ).
thf(func_def_62,type,
p: nat ).
thf(func_def_63,type,
q: nat ).
thf(func_def_77,type,
sK0: nat > ( nat > $o ) > nat ).
thf(func_def_78,type,
sK1: ( nat > $o ) > nat ).
thf(func_def_79,type,
sK2:
!>[X0: $tType,X1: $tType] : ( ( X1 > X0 ) > ( X1 > X0 ) > X1 ) ).
thf(func_def_80,type,
sK3: nat > nat > nat ).
thf(func_def_82,type,
ph5:
!>[X0: $tType] : X0 ).
thf(f1126,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f1125,f1105]) ).
thf(f1105,plain,
( ( genClo1015804716orrect @ p @ ( genClo1163638703lle_te @ p @ ( plus_plus @ nat @ i @ one_one @ nat ) ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f473]) ).
thf(f473,plain,
( ( genClo1015804716orrect @ p @ ( genClo1163638703lle_te @ p @ ( plus_plus @ nat @ i @ one_one @ nat ) ) )
= $true ),
inference(fool_elimination,[status(thm)],[f472]) ).
thf(f472,plain,
genClo1015804716orrect @ p @ ( genClo1163638703lle_te @ p @ ( plus_plus @ nat @ i @ one_one @ nat ) ),
inference(rectify,[status(thm)],[f2]) ).
thf(f2,axiom,
genClo1015804716orrect @ p @ ( genClo1163638703lle_te @ p @ ( plus_plus @ nat @ i @ one_one @ nat ) ),
file('/export/starexec/sandbox/tmp/tmp.XXNzzqTI4f/DTF2THF_9062.p',fact_1_corr__p) ).
thf(f1125,plain,
( ( genClo1015804716orrect @ p @ ( genClo1163638703lle_te @ p @ ( plus_plus @ nat @ i @ one_one @ nat ) ) )
!= $true ),
inference(subsumption_resolution,[status(thm)],[f1124,f1098]) ).
thf(f1098,plain,
( ( genClo1015804716orrect @ q @ ( genClo1163638703lle_te @ q @ ( plus_plus @ nat @ i @ one_one @ nat ) ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f629]) ).
thf(f629,plain,
( ( genClo1015804716orrect @ q @ ( genClo1163638703lle_te @ q @ ( plus_plus @ nat @ i @ one_one @ nat ) ) )
= $true ),
inference(fool_elimination,[status(thm)],[f628]) ).
thf(f628,plain,
genClo1015804716orrect @ q @ ( genClo1163638703lle_te @ q @ ( plus_plus @ nat @ i @ one_one @ nat ) ),
inference(rectify,[status(thm)],[f3]) ).
thf(f3,axiom,
genClo1015804716orrect @ q @ ( genClo1163638703lle_te @ q @ ( plus_plus @ nat @ i @ one_one @ nat ) ),
file('/export/starexec/sandbox/tmp/tmp.XXNzzqTI4f/DTF2THF_9062.p',fact_2_corr__q__tq) ).
thf(f1124,plain,
( ( ( genClo1015804716orrect @ q @ ( genClo1163638703lle_te @ q @ ( plus_plus @ nat @ i @ one_one @ nat ) ) )
!= $true )
| ( ( genClo1015804716orrect @ p @ ( genClo1163638703lle_te @ p @ ( plus_plus @ nat @ i @ one_one @ nat ) ) )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f1123]) ).
thf(f1123,plain,
( ( $true != $true )
| ( ( genClo1015804716orrect @ p @ ( genClo1163638703lle_te @ p @ ( plus_plus @ nat @ i @ one_one @ nat ) ) )
!= $true )
| ( ( genClo1015804716orrect @ q @ ( genClo1163638703lle_te @ q @ ( plus_plus @ nat @ i @ one_one @ nat ) ) )
!= $true ) ),
inference(superposition,[status(thm)],[f1085,f1091]) ).
thf(f1091,plain,
! [X2: nat,X0: nat,X1: nat] :
( ( $true
= ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( genClo1163638703lle_te @ X0 @ X2 ) @ ( genClo1163638703lle_te @ X1 @ X2 ) ) ) @ genClo1278781456e_beta ) )
| ( ( genClo1015804716orrect @ X1 @ ( genClo1163638703lle_te @ X1 @ X2 ) )
!= $true )
| ( $true
!= ( genClo1015804716orrect @ X0 @ ( genClo1163638703lle_te @ X0 @ X2 ) ) ) ),
inference(cnf_transformation,[status(thm)],[f998]) ).
thf(f998,plain,
! [X0: nat,X1: nat,X2: nat] :
( ( $true
!= ( genClo1015804716orrect @ X0 @ ( genClo1163638703lle_te @ X0 @ X2 ) ) )
| ( $true
= ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( genClo1163638703lle_te @ X0 @ X2 ) @ ( genClo1163638703lle_te @ X1 @ X2 ) ) ) @ genClo1278781456e_beta ) )
| ( ( genClo1015804716orrect @ X1 @ ( genClo1163638703lle_te @ X1 @ X2 ) )
!= $true ) ),
inference(flattening,[status(thm)],[f997]) ).
thf(f997,plain,
! [X0: nat,X1: nat,X2: nat] :
( ( $true
= ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( genClo1163638703lle_te @ X0 @ X2 ) @ ( genClo1163638703lle_te @ X1 @ X2 ) ) ) @ genClo1278781456e_beta ) )
| ( $true
!= ( genClo1015804716orrect @ X0 @ ( genClo1163638703lle_te @ X0 @ X2 ) ) )
| ( ( genClo1015804716orrect @ X1 @ ( genClo1163638703lle_te @ X1 @ X2 ) )
!= $true ) ),
inference(ennf_transformation,[status(thm)],[f549]) ).
thf(f549,plain,
! [X0: nat,X1: nat,X2: nat] :
( ( ( $true
= ( genClo1015804716orrect @ X0 @ ( genClo1163638703lle_te @ X0 @ X2 ) ) )
& ( ( genClo1015804716orrect @ X1 @ ( genClo1163638703lle_te @ X1 @ X2 ) )
= $true ) )
=> ( $true
= ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( genClo1163638703lle_te @ X0 @ X2 ) @ ( genClo1163638703lle_te @ X1 @ X2 ) ) ) @ genClo1278781456e_beta ) ) ),
inference(fool_elimination,[status(thm)],[f548]) ).
thf(f548,plain,
! [X0: nat,X1: nat,X2: nat] :
( ( ( genClo1015804716orrect @ X0 @ ( genClo1163638703lle_te @ X0 @ X2 ) )
& ( genClo1015804716orrect @ X1 @ ( genClo1163638703lle_te @ X1 @ X2 ) ) )
=> ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( genClo1163638703lle_te @ X0 @ X2 ) @ ( genClo1163638703lle_te @ X1 @ X2 ) ) ) @ genClo1278781456e_beta ) ),
inference(rectify,[status(thm)],[f1]) ).
thf(f1,axiom,
! [X1: nat,X2: nat,X3: nat] :
( ( ( genClo1015804716orrect @ X1 @ ( genClo1163638703lle_te @ X1 @ X3 ) )
& ( genClo1015804716orrect @ X2 @ ( genClo1163638703lle_te @ X2 @ X3 ) ) )
=> ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( genClo1163638703lle_te @ X1 @ X3 ) @ ( genClo1163638703lle_te @ X2 @ X3 ) ) ) @ genClo1278781456e_beta ) ),
file('/export/starexec/sandbox/tmp/tmp.XXNzzqTI4f/DTF2THF_9062.p',fact_0_rts2b) ).
thf(f1085,plain,
( ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( genClo1163638703lle_te @ p @ ( plus_plus @ nat @ i @ one_one @ nat ) ) @ ( genClo1163638703lle_te @ q @ ( plus_plus @ nat @ i @ one_one @ nat ) ) ) ) @ genClo1278781456e_beta )
!= $true ),
inference(cnf_transformation,[status(thm)],[f942]) ).
thf(f942,plain,
( ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( genClo1163638703lle_te @ p @ ( plus_plus @ nat @ i @ one_one @ nat ) ) @ ( genClo1163638703lle_te @ q @ ( plus_plus @ nat @ i @ one_one @ nat ) ) ) ) @ genClo1278781456e_beta )
!= $true ),
inference(flattening,[status(thm)],[f403]) ).
thf(f403,plain,
( ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( genClo1163638703lle_te @ p @ ( plus_plus @ nat @ i @ one_one @ nat ) ) @ ( genClo1163638703lle_te @ q @ ( plus_plus @ nat @ i @ one_one @ nat ) ) ) ) @ genClo1278781456e_beta )
!= $true ),
inference(fool_elimination,[status(thm)],[f402]) ).
thf(f402,plain,
~ ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( genClo1163638703lle_te @ p @ ( plus_plus @ nat @ i @ one_one @ nat ) ) @ ( genClo1163638703lle_te @ q @ ( plus_plus @ nat @ i @ one_one @ nat ) ) ) ) @ genClo1278781456e_beta ),
inference(rectify,[status(thm)],[f326]) ).
thf(f326,negated_conjecture,
~ ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( genClo1163638703lle_te @ p @ ( plus_plus @ nat @ i @ one_one @ nat ) ) @ ( genClo1163638703lle_te @ q @ ( plus_plus @ nat @ i @ one_one @ nat ) ) ) ) @ genClo1278781456e_beta ),
inference(negated_conjecture,[status(cth)],[f325]) ).
thf(f325,conjecture,
ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( genClo1163638703lle_te @ p @ ( plus_plus @ nat @ i @ one_one @ nat ) ) @ ( genClo1163638703lle_te @ q @ ( plus_plus @ nat @ i @ one_one @ nat ) ) ) ) @ genClo1278781456e_beta,
file('/export/starexec/sandbox/tmp/tmp.XXNzzqTI4f/DTF2THF_9062.p',conj_0) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP063^2 : TPTP v9.2.1. Released v7.5.0.
% 0.12/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.16/0.34 % Computer : n002.cluster.edu
% 0.16/0.34 % Model : x86_64 x86_64
% 0.16/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.34 % Memory : 8042.1875MB
% 0.16/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.34 % CPULimit : 300
% 0.16/0.34 % WCLimit : 300
% 0.16/0.34 % DateTime : Tue Feb 24 08:25:33 EST 2026
% 0.16/0.34 % CPUTime :
% 0.16/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.28/0.46 ---- Original DTF file ---
% 0.28/0.46 thf(spec,logic,$$dhol).
% 0.28/0.46 %------------------------------------------------------------------------------
% 0.28/0.46 % File : ITP063^2 : TPTP v9.2.1. Released v7.5.0.
% 0.28/0.46 % Domain : Interactive Theorem Proving
% 0.28/0.46 % Problem : Sledgehammer GenClock problem prob_539__3247850_1
% 0.28/0.46 % Version : Especial.
% 0.28/0.46 % English :
% 0.28/0.46
% 0.28/0.46 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.28/0.46 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.28/0.46 % Source : [Des21]
% 0.28/0.46 % Names : GenClock/prob_539__3247850_1 [Des21]
% 0.28/0.46
% 0.28/0.46 % Status : Theorem
% 0.28/0.46 % Rating : 0.33 v9.1.0, 0.00 v7.5.0
% 0.28/0.46 % Syntax : Number of formulae : 389 ( 110 unt; 64 typ; 0 def)
% 0.28/0.46 % Number of atoms : 780 ( 294 equ; 0 cnn)
% 0.28/0.46 % Maximal formula atoms : 6 ( 2 avg)
% 0.28/0.46 % Number of connectives : 3102 ( 14 ~; 9 |; 20 &;2760 @)
% 0.28/0.46 % ( 0 <=>; 299 =>; 0 <=; 0 <~>)
% 0.28/0.46 % Maximal formula depth : 16 ( 6 avg)
% 0.28/0.46 % Number of types : 3 ( 2 usr)
% 0.28/0.46 % Number of type conns : 32 ( 32 >; 0 *; 0 +; 0 <<)
% 0.28/0.46 % Number of symbols : 63 ( 62 usr; 9 con; 0-4 aty)
% 0.28/0.46 % Number of variables : 786 ( 21 ^; 712 !; 4 ?; 786 :)
% 0.28/0.46 % ( 49 !>; 0 ?*; 0 @-; 0 @+)
% 0.28/0.46 % SPC : TH1_THM_EQU_NAR
% 0.28/0.46
% 0.28/0.46 % Comments : This file was generated by Sledgehammer 2021-02-23 16:16:06.947
% 0.28/0.46 %------------------------------------------------------------------------------
% 0.28/0.46 % Could-be-implicit typings (2)
% 0.28/0.46 thf(ty_t_Real_Oreal,type,
% 0.28/0.46 real: $tType ).
% 0.28/0.46
% 0.28/0.46 thf(ty_t_Nat_Onat,type,
% 0.28/0.46 nat: $tType ).
% 0.28/0.46
% 0.28/0.46 % Explicit typings (62)
% 0.28/0.46 thf(sy_cl_HOL_Otype,type,
% 0.28/0.46 type:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Groups_Oone,type,
% 0.28/0.46 one:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Rings_Oring,type,
% 0.28/0.46 ring:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Groups_Ozero,type,
% 0.28/0.46 zero:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Groups_Ominus,type,
% 0.28/0.46 minus:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Rings_Osemiring,type,
% 0.28/0.46 semiring:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Num_Oneg__numeral,type,
% 0.28/0.46 neg_numeral:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Rings_Omult__zero,type,
% 0.28/0.46 mult_zero:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Groups_Ogroup__add,type,
% 0.28/0.46 group_add:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Groups_Omonoid__add,type,
% 0.28/0.46 monoid_add:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Groups_Omonoid__mult,type,
% 0.28/0.46 monoid_mult:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Rings_Oidom__abs__sgn,type,
% 0.28/0.46 idom_abs_sgn:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Rings_Ozero__neq__one,type,
% 0.28/0.46 zero_neq_one:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Groups_Oab__group__add,type,
% 0.28/0.46 ab_group_add:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Rings_Ocomm__semiring,type,
% 0.28/0.46 comm_semiring:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Groups_Osemigroup__add,type,
% 0.28/0.46 semigroup_add:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Groups_Osemigroup__mult,type,
% 0.28/0.46 semigroup_mult:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Rings_Olinordered__idom,type,
% 0.28/0.46 linordered_idom:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Groups_Ocomm__monoid__add,type,
% 0.28/0.46 comm_monoid_add:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Rings_Oordered__semiring,type,
% 0.28/0.46 ordered_semiring:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Groups_Oab__semigroup__add,type,
% 0.28/0.46 ab_semigroup_add:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Groups_Ocomm__monoid__diff,type,
% 0.28/0.46 comm_monoid_diff:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Groups_Ocomm__monoid__mult,type,
% 0.28/0.46 comm_monoid_mult:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Groups_Oab__semigroup__mult,type,
% 0.28/0.46 ab_semigroup_mult:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Rings_Olinordered__semidom,type,
% 0.28/0.46 linordered_semidom:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Groups_Ocancel__semigroup__add,type,
% 0.28/0.46 cancel_semigroup_add:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 0.28/0.46 ordered_ab_group_add:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Rings_Ocomm__semiring__1__cancel,type,
% 0.28/0.46 comm_s1003936772cancel:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
% 0.28/0.46 cancel1352612707id_add:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Rings_Oring__1__no__zero__divisors,type,
% 0.28/0.46 ring_11004092258visors:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
% 0.28/0.46 cancel146912293up_add:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Groups_Olinordered__ab__group__add,type,
% 0.28/0.46 linord219039673up_add:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Groups_Oordered__comm__monoid__add,type,
% 0.28/0.46 ordere216010020id_add:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Groups_Oordered__ab__group__add__abs,type,
% 0.28/0.46 ordere142940540dd_abs:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
% 0.28/0.46 ordere779506340up_add:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Rings_Osemiring__no__zero__divisors,type,
% 0.28/0.46 semiri1193490041visors:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
% 0.28/0.46 linord1659791738miring:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 0.28/0.46 canoni770627133id_add:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
% 0.28/0.46 ordere236663937imp_le:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Groups_Oordered__cancel__comm__monoid__diff,type,
% 0.28/0.46 ordere623563068d_diff:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Rings_Osemiring__no__zero__divisors__cancel,type,
% 0.28/0.46 semiri1923998003cancel:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Groups_Oordered__ab__semigroup__monoid__add__imp__le,type,
% 0.28/0.46 ordere516151231imp_le:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_GenClock__Mirabelle__pukziqipvs_OPC,type,
% 0.28/0.46 genClo1161277105lle_PC: nat > real > real ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_GenClock__Mirabelle__pukziqipvs_O_092_060beta_062,type,
% 0.28/0.46 genClo1278781456e_beta: real ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_GenClock__Mirabelle__pukziqipvs_O_092_060mu_062,type,
% 0.28/0.46 genClo872980712lle_mu: real ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_GenClock__Mirabelle__pukziqipvs_O_092_060rho_062,type,
% 0.28/0.46 genClo1144207539le_rho: real ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_GenClock__Mirabelle__pukziqipvs_Ocorrect,type,
% 0.28/0.46 genClo1015804716orrect: nat > real > $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_GenClock__Mirabelle__pukziqipvs_OokRead2,type,
% 0.28/0.46 genClo293725282kRead2: ( nat > real ) > ( nat > real ) > real > ( nat > $o ) > $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_GenClock__Mirabelle__pukziqipvs_Ormax,type,
% 0.28/0.46 genClo1650508560e_rmax: real ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_GenClock__Mirabelle__pukziqipvs_Ormin,type,
% 0.28/0.46 genClo1651033342e_rmin: real ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_GenClock__Mirabelle__pukziqipvs_Ote,type,
% 0.28/0.46 genClo1163638703lle_te: nat > nat > real ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Groups_Oabs__class_Oabs,type,
% 0.28/0.46 abs_abs:
% 0.28/0.46 !>[A: $tType] : ( A > A ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Groups_Ominus__class_Ominus,type,
% 0.28/0.46 minus_minus:
% 0.28/0.46 !>[A: $tType] : ( A > A > A ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Groups_Oone__class_Oone,type,
% 0.28/0.46 one_one:
% 0.28/0.46 !>[A: $tType] : A ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Groups_Oplus__class_Oplus,type,
% 0.28/0.46 plus_plus:
% 0.28/0.46 !>[A: $tType] : ( A > A > A ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Groups_Otimes__class_Otimes,type,
% 0.28/0.46 times_times:
% 0.28/0.46 !>[A: $tType] : ( A > A > A ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 0.28/0.46 zero_zero:
% 0.28/0.46 !>[A: $tType] : A ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.28/0.46 ord_less_eq:
% 0.28/0.46 !>[A: $tType] : ( A > A > $o ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_v_i,type,
% 0.28/0.46 i: nat ).
% 0.28/0.46
% 0.28/0.46 thf(sy_v_l,type,
% 0.28/0.46 l: nat ).
% 0.28/0.46
% 0.28/0.46 thf(sy_v_p,type,
% 0.28/0.46 p: nat ).
% 0.28/0.46
% 0.28/0.46 thf(sy_v_q,type,
% 0.28/0.46 q: nat ).
% 0.28/0.46
% 0.28/0.46 % Relevant facts (253)
% 0.28/0.46 thf(fact_0_rts2b,axiom,
% 0.28/0.46 ! [P: nat,Q: nat,I: nat] :
% 0.28/0.46 ( ( ( genClo1015804716orrect @ P @ ( genClo1163638703lle_te @ P @ I ) )
% 0.28/0.46 & ( genClo1015804716orrect @ Q @ ( genClo1163638703lle_te @ Q @ I ) ) )
% 0.28/0.46 => ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( genClo1163638703lle_te @ P @ I ) @ ( genClo1163638703lle_te @ Q @ I ) ) ) @ genClo1278781456e_beta ) ) ).
% 0.28/0.46
% 0.28/0.46 % rts2b
% 0.28/0.46 thf(fact_1_corr__p,axiom,
% 0.28/0.46 genClo1015804716orrect @ p @ ( genClo1163638703lle_te @ p @ ( plus_plus @ nat @ i @ ( one_one @ nat ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % corr_p
% 0.28/0.46 thf(fact_2_corr__q__tq,axiom,
% 0.28/0.46 genClo1015804716orrect @ q @ ( genClo1163638703lle_te @ q @ ( plus_plus @ nat @ i @ ( one_one @ nat ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % corr_q_tq
% 0.28/0.46 thf(fact_3_ie,axiom,
% 0.28/0.46 ord_less_eq @ real @ ( genClo1163638703lle_te @ q @ ( plus_plus @ nat @ i @ ( one_one @ nat ) ) ) @ ( genClo1163638703lle_te @ p @ ( plus_plus @ nat @ i @ ( one_one @ nat ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % ie
% 0.28/0.46 thf(fact_4_corr__q,axiom,
% 0.28/0.46 genClo1015804716orrect @ q @ ( genClo1163638703lle_te @ p @ ( plus_plus @ nat @ i @ ( one_one @ nat ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % corr_q
% 0.28/0.46 thf(fact_5_corr__l,axiom,
% 0.28/0.46 genClo1015804716orrect @ l @ ( genClo1163638703lle_te @ p @ ( plus_plus @ nat @ i @ ( one_one @ nat ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % corr_l
% 0.28/0.46 thf(fact_6_posD,axiom,
% 0.28/0.46 ord_less_eq @ real @ ( zero_zero @ real ) @ ( minus_minus @ real @ ( genClo1163638703lle_te @ p @ ( plus_plus @ nat @ i @ ( one_one @ nat ) ) ) @ ( genClo1163638703lle_te @ q @ ( plus_plus @ nat @ i @ ( one_one @ nat ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % posD
% 0.28/0.46 thf(fact_7_corr__l__tq,axiom,
% 0.28/0.46 genClo1015804716orrect @ l @ ( genClo1163638703lle_te @ q @ ( plus_plus @ nat @ i @ ( one_one @ nat ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % corr_l_tq
% 0.28/0.46 thf(fact_8_le__add__diff__inverse,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( linordered_semidom @ A )
% 0.28/0.46 => ! [B: A,A2: A] :
% 0.28/0.46 ( ( ord_less_eq @ A @ B @ A2 )
% 0.28/0.46 => ( ( plus_plus @ A @ B @ ( minus_minus @ A @ A2 @ B ) )
% 0.28/0.46 = A2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % le_add_diff_inverse
% 0.28/0.46 thf(fact_9_le__add__diff__inverse2,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( linordered_semidom @ A )
% 0.28/0.46 => ! [B: A,A2: A] :
% 0.28/0.46 ( ( ord_less_eq @ A @ B @ A2 )
% 0.28/0.46 => ( ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B ) @ B )
% 0.28/0.46 = A2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % le_add_diff_inverse2
% 0.28/0.46 thf(fact_10_rte,axiom,
% 0.28/0.46 ! [P2: nat,I2: nat] :
% 0.28/0.46 ( ( genClo1015804716orrect @ P2 @ ( genClo1163638703lle_te @ P2 @ ( plus_plus @ nat @ I2 @ ( one_one @ nat ) ) ) )
% 0.28/0.46 => ( ord_less_eq @ real @ ( genClo1163638703lle_te @ P2 @ I2 ) @ ( genClo1163638703lle_te @ P2 @ ( plus_plus @ nat @ I2 @ ( one_one @ nat ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % rte
% 0.28/0.46 thf(fact_11_abs__1,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( idom_abs_sgn @ A )
% 0.28/0.46 => ( ( abs_abs @ A @ ( one_one @ A ) )
% 0.28/0.46 = ( one_one @ A ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % abs_1
% 0.28/0.46 thf(fact_12_abs__add__abs,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( ordere142940540dd_abs @ A )
% 0.28/0.46 => ! [A2: A,B: A] :
% 0.28/0.46 ( ( abs_abs @ A @ ( plus_plus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B ) ) )
% 0.28/0.46 = ( plus_plus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % abs_add_abs
% 0.28/0.46 thf(fact_13_add__diff__cancel,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( group_add @ A )
% 0.28/0.46 => ! [A2: A,B: A] :
% 0.28/0.46 ( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B ) @ B )
% 0.28/0.46 = A2 ) ) ).
% 0.28/0.46
% 0.28/0.46 % add_diff_cancel
% 0.28/0.46 thf(fact_14_diff__add__cancel,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( group_add @ A )
% 0.28/0.46 => ! [A2: A,B: A] :
% 0.28/0.46 ( ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B ) @ B )
% 0.28/0.46 = A2 ) ) ).
% 0.28/0.46
% 0.28/0.46 % diff_add_cancel
% 0.28/0.46 thf(fact_15_add__diff__cancel__left,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( cancel146912293up_add @ A )
% 0.28/0.46 => ! [C: A,A2: A,B: A] :
% 0.28/0.46 ( ( minus_minus @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) )
% 0.28/0.46 = ( minus_minus @ A @ A2 @ B ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % add_diff_cancel_left
% 0.28/0.46 thf(fact_16_add__diff__cancel__left_H,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( cancel146912293up_add @ A )
% 0.28/0.46 => ! [A2: A,B: A] :
% 0.28/0.46 ( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B ) @ A2 )
% 0.28/0.46 = B ) ) ).
% 0.28/0.46
% 0.28/0.46 % add_diff_cancel_left'
% 0.28/0.46 thf(fact_17_add__diff__cancel__right,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( cancel146912293up_add @ A )
% 0.28/0.46 => ! [A2: A,C: A,B: A] :
% 0.28/0.46 ( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) )
% 0.28/0.46 = ( minus_minus @ A @ A2 @ B ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % add_diff_cancel_right
% 0.28/0.46 thf(fact_18_add__right__cancel,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( cancel_semigroup_add @ A )
% 0.28/0.46 => ! [B: A,A2: A,C: A] :
% 0.28/0.46 ( ( ( plus_plus @ A @ B @ A2 )
% 0.28/0.46 = ( plus_plus @ A @ C @ A2 ) )
% 0.28/0.46 = ( B = C ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % add_right_cancel
% 0.28/0.46 thf(fact_19_add__left__cancel,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( cancel_semigroup_add @ A )
% 0.28/0.46 => ! [A2: A,B: A,C: A] :
% 0.28/0.46 ( ( ( plus_plus @ A @ A2 @ B )
% 0.28/0.46 = ( plus_plus @ A @ A2 @ C ) )
% 0.28/0.46 = ( B = C ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % add_left_cancel
% 0.28/0.46 thf(fact_20_abs__idempotent,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( ordere142940540dd_abs @ A )
% 0.28/0.46 => ! [A2: A] :
% 0.28/0.46 ( ( abs_abs @ A @ ( abs_abs @ A @ A2 ) )
% 0.28/0.46 = ( abs_abs @ A @ A2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % abs_idempotent
% 0.28/0.46 thf(fact_21_abs__abs,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( idom_abs_sgn @ A )
% 0.28/0.46 => ! [A2: A] :
% 0.28/0.46 ( ( abs_abs @ A @ ( abs_abs @ A @ A2 ) )
% 0.28/0.46 = ( abs_abs @ A @ A2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % abs_abs
% 0.28/0.46 thf(fact_22_le__zero__eq,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( canoni770627133id_add @ A )
% 0.28/0.46 => ! [N: A] :
% 0.28/0.46 ( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
% 0.28/0.46 = ( N
% 0.28/0.46 = ( zero_zero @ A ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % le_zero_eq
% 0.28/0.46 thf(fact_23_add__le__cancel__right,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( ordere236663937imp_le @ A )
% 0.28/0.46 => ! [A2: A,C: A,B: A] :
% 0.28/0.46 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) )
% 0.28/0.46 = ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % add_le_cancel_right
% 0.28/0.46 thf(fact_24_add__le__cancel__left,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( ordere236663937imp_le @ A )
% 0.28/0.46 => ! [C: A,A2: A,B: A] :
% 0.28/0.46 ( ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) )
% 0.28/0.46 = ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % add_le_cancel_left
% 0.28/0.46 thf(fact_25_zero__eq__add__iff__both__eq__0,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( canoni770627133id_add @ A )
% 0.28/0.46 => ! [X: A,Y: A] :
% 0.28/0.46 ( ( ( zero_zero @ A )
% 0.28/0.46 = ( plus_plus @ A @ X @ Y ) )
% 0.28/0.46 = ( ( X
% 0.28/0.46 = ( zero_zero @ A ) )
% 0.28/0.46 & ( Y
% 0.28/0.46 = ( zero_zero @ A ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % zero_eq_add_iff_both_eq_0
% 0.28/0.46 thf(fact_26_add__eq__0__iff__both__eq__0,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( canoni770627133id_add @ A )
% 0.28/0.46 => ! [X: A,Y: A] :
% 0.28/0.46 ( ( ( plus_plus @ A @ X @ Y )
% 0.28/0.46 = ( zero_zero @ A ) )
% 0.28/0.46 = ( ( X
% 0.28/0.46 = ( zero_zero @ A ) )
% 0.28/0.46 & ( Y
% 0.28/0.46 = ( zero_zero @ A ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % add_eq_0_iff_both_eq_0
% 0.28/0.46 thf(fact_27_add__cancel__right__right,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( cancel1352612707id_add @ A )
% 0.28/0.46 => ! [A2: A,B: A] :
% 0.28/0.46 ( ( A2
% 0.28/0.46 = ( plus_plus @ A @ A2 @ B ) )
% 0.28/0.46 = ( B
% 0.28/0.46 = ( zero_zero @ A ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % add_cancel_right_right
% 0.28/0.46 thf(fact_28_add__cancel__right__left,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( cancel1352612707id_add @ A )
% 0.28/0.46 => ! [A2: A,B: A] :
% 0.28/0.46 ( ( A2
% 0.28/0.46 = ( plus_plus @ A @ B @ A2 ) )
% 0.28/0.46 = ( B
% 0.28/0.46 = ( zero_zero @ A ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % add_cancel_right_left
% 0.28/0.46 thf(fact_29_add__cancel__left__right,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( cancel1352612707id_add @ A )
% 0.28/0.46 => ! [A2: A,B: A] :
% 0.28/0.46 ( ( ( plus_plus @ A @ A2 @ B )
% 0.28/0.46 = A2 )
% 0.28/0.46 = ( B
% 0.28/0.46 = ( zero_zero @ A ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % add_cancel_left_right
% 0.28/0.46 thf(fact_30_add__cancel__left__left,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( cancel1352612707id_add @ A )
% 0.28/0.46 => ! [B: A,A2: A] :
% 0.28/0.46 ( ( ( plus_plus @ A @ B @ A2 )
% 0.28/0.46 = A2 )
% 0.28/0.46 = ( B
% 0.28/0.46 = ( zero_zero @ A ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % add_cancel_left_left
% 0.28/0.46 thf(fact_31_double__zero__sym,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( linord219039673up_add @ A )
% 0.28/0.46 => ! [A2: A] :
% 0.28/0.46 ( ( ( zero_zero @ A )
% 0.28/0.46 = ( plus_plus @ A @ A2 @ A2 ) )
% 0.28/0.46 = ( A2
% 0.28/0.46 = ( zero_zero @ A ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % double_zero_sym
% 0.28/0.46 thf(fact_32_double__zero,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( linord219039673up_add @ A )
% 0.28/0.46 => ! [A2: A] :
% 0.28/0.46 ( ( ( plus_plus @ A @ A2 @ A2 )
% 0.28/0.46 = ( zero_zero @ A ) )
% 0.28/0.46 = ( A2
% 0.28/0.46 = ( zero_zero @ A ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % double_zero
% 0.28/0.46 thf(fact_33_add_Oright__neutral,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( monoid_add @ A )
% 0.28/0.46 => ! [A2: A] :
% 0.28/0.46 ( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
% 0.28/0.46 = A2 ) ) ).
% 0.28/0.46
% 0.28/0.46 % add.right_neutral
% 0.28/0.46 thf(fact_34_add_Oleft__neutral,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( monoid_add @ A )
% 0.28/0.46 => ! [A2: A] :
% 0.28/0.46 ( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
% 0.28/0.46 = A2 ) ) ).
% 0.28/0.46
% 0.28/0.46 % add.left_neutral
% 0.28/0.46 thf(fact_35_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( cancel1352612707id_add @ A )
% 0.28/0.46 => ! [A2: A] :
% 0.28/0.46 ( ( minus_minus @ A @ A2 @ A2 )
% 0.28/0.46 = ( zero_zero @ A ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % cancel_comm_monoid_add_class.diff_cancel
% 0.28/0.46 thf(fact_36_diff__zero,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( cancel1352612707id_add @ A )
% 0.28/0.46 => ! [A2: A] :
% 0.28/0.46 ( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
% 0.28/0.46 = A2 ) ) ).
% 0.28/0.46
% 0.28/0.46 % diff_zero
% 0.28/0.46 thf(fact_37_zero__diff,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( comm_monoid_diff @ A )
% 0.28/0.46 => ! [A2: A] :
% 0.28/0.46 ( ( minus_minus @ A @ ( zero_zero @ A ) @ A2 )
% 0.28/0.46 = ( zero_zero @ A ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % zero_diff
% 0.28/0.46 thf(fact_38_diff__0__right,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( group_add @ A )
% 0.28/0.46 => ! [A2: A] :
% 0.28/0.46 ( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
% 0.28/0.46 = A2 ) ) ).
% 0.28/0.46
% 0.28/0.46 % diff_0_right
% 0.28/0.46 thf(fact_39_diff__self,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( group_add @ A )
% 0.28/0.46 => ! [A2: A] :
% 0.28/0.46 ( ( minus_minus @ A @ A2 @ A2 )
% 0.28/0.46 = ( zero_zero @ A ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % diff_self
% 0.28/0.46 thf(fact_40_add__diff__cancel__right_H,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( cancel146912293up_add @ A )
% 0.28/0.46 => ! [A2: A,B: A] :
% 0.28/0.46 ( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B ) @ B )
% 0.28/0.46 = A2 ) ) ).
% 0.28/0.46
% 0.28/0.46 % add_diff_cancel_right'
% 0.28/0.47 thf(fact_41_abs__zero,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ordere142940540dd_abs @ A )
% 0.28/0.47 => ( ( abs_abs @ A @ ( zero_zero @ A ) )
% 0.28/0.47 = ( zero_zero @ A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % abs_zero
% 0.28/0.47 thf(fact_42_abs__eq__0,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ordere142940540dd_abs @ A )
% 0.28/0.47 => ! [A2: A] :
% 0.28/0.47 ( ( ( abs_abs @ A @ A2 )
% 0.28/0.47 = ( zero_zero @ A ) )
% 0.28/0.47 = ( A2
% 0.28/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % abs_eq_0
% 0.28/0.47 thf(fact_43_ext,axiom,
% 0.28/0.47 ! [B2: $tType,A: $tType,F: A > B2,G: A > B2] :
% 0.28/0.47 ( ! [X2: A] :
% 0.28/0.47 ( ( F @ X2 )
% 0.28/0.47 = ( G @ X2 ) )
% 0.28/0.47 => ( F = G ) ) ).
% 0.28/0.47
% 0.28/0.47 % ext
% 0.28/0.47 thf(fact_44_abs__0__eq,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ordere142940540dd_abs @ A )
% 0.28/0.47 => ! [A2: A] :
% 0.28/0.47 ( ( ( zero_zero @ A )
% 0.28/0.47 = ( abs_abs @ A @ A2 ) )
% 0.28/0.47 = ( A2
% 0.28/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % abs_0_eq
% 0.28/0.47 thf(fact_45_abs__0,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( idom_abs_sgn @ A )
% 0.28/0.47 => ( ( abs_abs @ A @ ( zero_zero @ A ) )
% 0.28/0.47 = ( zero_zero @ A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % abs_0
% 0.28/0.47 thf(fact_46_zero__le__double__add__iff__zero__le__single__add,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linord219039673up_add @ A )
% 0.28/0.47 => ! [A2: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ A2 ) )
% 0.28/0.47 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % zero_le_double_add_iff_zero_le_single_add
% 0.28/0.47 thf(fact_47_double__add__le__zero__iff__single__add__le__zero,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linord219039673up_add @ A )
% 0.28/0.47 => ! [A2: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ A2 ) @ ( zero_zero @ A ) )
% 0.28/0.47 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % double_add_le_zero_iff_single_add_le_zero
% 0.28/0.47 thf(fact_48_le__add__same__cancel2,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ordere516151231imp_le @ A )
% 0.28/0.47 => ! [A2: A,B: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ B @ A2 ) )
% 0.28/0.47 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ B ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % le_add_same_cancel2
% 0.28/0.47 thf(fact_49_le__add__same__cancel1,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ordere516151231imp_le @ A )
% 0.28/0.47 => ! [A2: A,B: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ A2 @ B ) )
% 0.28/0.47 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ B ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % le_add_same_cancel1
% 0.28/0.47 thf(fact_50_add__le__same__cancel2,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ordere516151231imp_le @ A )
% 0.28/0.47 => ! [A2: A,B: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B ) @ B )
% 0.28/0.47 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % add_le_same_cancel2
% 0.28/0.47 thf(fact_51_add__le__same__cancel1,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ordere516151231imp_le @ A )
% 0.28/0.47 => ! [B: A,A2: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ ( plus_plus @ A @ B @ A2 ) @ B )
% 0.28/0.47 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % add_le_same_cancel1
% 0.28/0.47 thf(fact_52_diff__ge__0__iff__ge,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ordered_ab_group_add @ A )
% 0.28/0.47 => ! [A2: A,B: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A2 @ B ) )
% 0.28/0.47 = ( ord_less_eq @ A @ B @ A2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % diff_ge_0_iff_ge
% 0.28/0.47 thf(fact_53_diff__add__zero,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( comm_monoid_diff @ A )
% 0.28/0.47 => ! [A2: A,B: A] :
% 0.28/0.47 ( ( minus_minus @ A @ A2 @ ( plus_plus @ A @ A2 @ B ) )
% 0.28/0.47 = ( zero_zero @ A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % diff_add_zero
% 0.28/0.47 thf(fact_54_abs__le__zero__iff,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ordere142940540dd_abs @ A )
% 0.28/0.47 => ! [A2: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ ( zero_zero @ A ) )
% 0.28/0.47 = ( A2
% 0.28/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % abs_le_zero_iff
% 0.28/0.47 thf(fact_55_abs__le__self__iff,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ordere142940540dd_abs @ A )
% 0.28/0.47 => ! [A2: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ A2 )
% 0.28/0.47 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % abs_le_self_iff
% 0.28/0.47 thf(fact_56_abs__of__nonneg,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ordere142940540dd_abs @ A )
% 0.28/0.47 => ! [A2: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 0.28/0.47 => ( ( abs_abs @ A @ A2 )
% 0.28/0.47 = A2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % abs_of_nonneg
% 0.28/0.47 thf(fact_57_zero__reorient,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( zero @ A )
% 0.28/0.47 => ! [X: A] :
% 0.28/0.47 ( ( ( zero_zero @ A )
% 0.28/0.47 = X )
% 0.28/0.47 = ( X
% 0.28/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % zero_reorient
% 0.28/0.47 thf(fact_58_zero__le,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( canoni770627133id_add @ A )
% 0.28/0.47 => ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% 0.28/0.47
% 0.28/0.47 % zero_le
% 0.28/0.47 thf(fact_59_add_Ogroup__left__neutral,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( group_add @ A )
% 0.28/0.47 => ! [A2: A] :
% 0.28/0.47 ( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
% 0.28/0.47 = A2 ) ) ).
% 0.28/0.47
% 0.28/0.47 % add.group_left_neutral
% 0.28/0.47 thf(fact_60_add_Ocomm__neutral,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( comm_monoid_add @ A )
% 0.28/0.47 => ! [A2: A] :
% 0.28/0.47 ( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
% 0.28/0.47 = A2 ) ) ).
% 0.28/0.47
% 0.28/0.47 % add.comm_neutral
% 0.28/0.47 thf(fact_61_comm__monoid__add__class_Oadd__0,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( comm_monoid_add @ A )
% 0.28/0.47 => ! [A2: A] :
% 0.28/0.47 ( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
% 0.28/0.47 = A2 ) ) ).
% 0.28/0.47
% 0.28/0.47 % comm_monoid_add_class.add_0
% 0.28/0.47 thf(fact_62_zero__neq__one,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( zero_neq_one @ A )
% 0.28/0.47 => ( ( zero_zero @ A )
% 0.28/0.47 != ( one_one @ A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % zero_neq_one
% 0.28/0.47 thf(fact_63_eq__iff__diff__eq__0,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( group_add @ A )
% 0.28/0.47 => ( ( ^ [Y2: A,Z: A] : ( Y2 = Z ) )
% 0.28/0.47 = ( ^ [A3: A,B3: A] :
% 0.28/0.47 ( ( minus_minus @ A @ A3 @ B3 )
% 0.28/0.47 = ( zero_zero @ A ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % eq_iff_diff_eq_0
% 0.28/0.47 thf(fact_64_abs__eq__0__iff,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( idom_abs_sgn @ A )
% 0.28/0.47 => ! [A2: A] :
% 0.28/0.47 ( ( ( abs_abs @ A @ A2 )
% 0.28/0.47 = ( zero_zero @ A ) )
% 0.28/0.47 = ( A2
% 0.28/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % abs_eq_0_iff
% 0.28/0.47 thf(fact_65_add__nonpos__eq__0__iff,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ordere216010020id_add @ A )
% 0.28/0.47 => ! [X: A,Y: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
% 0.28/0.47 => ( ( ord_less_eq @ A @ Y @ ( zero_zero @ A ) )
% 0.28/0.47 => ( ( ( plus_plus @ A @ X @ Y )
% 0.28/0.47 = ( zero_zero @ A ) )
% 0.28/0.47 = ( ( X
% 0.28/0.47 = ( zero_zero @ A ) )
% 0.28/0.47 & ( Y
% 0.28/0.47 = ( zero_zero @ A ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % add_nonpos_eq_0_iff
% 0.28/0.47 thf(fact_66_add__nonneg__eq__0__iff,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ordere216010020id_add @ A )
% 0.28/0.47 => ! [X: A,Y: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 0.28/0.47 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
% 0.28/0.47 => ( ( ( plus_plus @ A @ X @ Y )
% 0.28/0.47 = ( zero_zero @ A ) )
% 0.28/0.47 = ( ( X
% 0.28/0.47 = ( zero_zero @ A ) )
% 0.28/0.47 & ( Y
% 0.28/0.47 = ( zero_zero @ A ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % add_nonneg_eq_0_iff
% 0.28/0.47 thf(fact_67_add__nonpos__nonpos,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ordere216010020id_add @ A )
% 0.28/0.47 => ! [A2: A,B: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
% 0.28/0.47 => ( ( ord_less_eq @ A @ B @ ( zero_zero @ A ) )
% 0.28/0.47 => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % add_nonpos_nonpos
% 0.28/0.47 thf(fact_68_add__nonneg__nonneg,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ordere216010020id_add @ A )
% 0.28/0.47 => ! [A2: A,B: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 0.28/0.47 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B )
% 0.28/0.47 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % add_nonneg_nonneg
% 0.28/0.47 thf(fact_69_add__increasing2,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ordere216010020id_add @ A )
% 0.28/0.47 => ! [C: A,B: A,A2: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C )
% 0.28/0.47 => ( ( ord_less_eq @ A @ B @ A2 )
% 0.28/0.47 => ( ord_less_eq @ A @ B @ ( plus_plus @ A @ A2 @ C ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % add_increasing2
% 0.28/0.47 thf(fact_70_add__decreasing2,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ordere216010020id_add @ A )
% 0.28/0.47 => ! [C: A,A2: A,B: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ C @ ( zero_zero @ A ) )
% 0.28/0.47 => ( ( ord_less_eq @ A @ A2 @ B )
% 0.28/0.47 => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % add_decreasing2
% 0.28/0.47 thf(fact_71_add__increasing,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ordere216010020id_add @ A )
% 0.28/0.47 => ! [A2: A,B: A,C: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 0.28/0.47 => ( ( ord_less_eq @ A @ B @ C )
% 0.28/0.47 => ( ord_less_eq @ A @ B @ ( plus_plus @ A @ A2 @ C ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % add_increasing
% 0.28/0.47 thf(fact_72_add__decreasing,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ordere216010020id_add @ A )
% 0.28/0.47 => ! [A2: A,C: A,B: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
% 0.28/0.47 => ( ( ord_less_eq @ A @ C @ B )
% 0.28/0.47 => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % add_decreasing
% 0.28/0.47 thf(fact_73_not__one__le__zero,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linord1659791738miring @ A )
% 0.28/0.47 => ~ ( ord_less_eq @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % not_one_le_zero
% 0.28/0.47 thf(fact_74_zero__le__one,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linord1659791738miring @ A )
% 0.28/0.47 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % zero_le_one
% 0.28/0.47 thf(fact_75_le__iff__diff__le__0,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ordered_ab_group_add @ A )
% 0.28/0.47 => ( ( ord_less_eq @ A )
% 0.28/0.47 = ( ^ [A3: A,B3: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ B3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % le_iff_diff_le_0
% 0.28/0.47 thf(fact_76_abs__ge__zero,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ordere142940540dd_abs @ A )
% 0.28/0.47 => ! [A2: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( abs_abs @ A @ A2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % abs_ge_zero
% 0.28/0.47 thf(fact_77_rts2a,axiom,
% 0.28/0.47 ! [P: nat,Q: nat,T: real,I: nat] :
% 0.28/0.47 ( ( ( genClo1015804716orrect @ P @ T )
% 0.28/0.47 & ( genClo1015804716orrect @ Q @ T )
% 0.28/0.47 & ( ord_less_eq @ real @ ( plus_plus @ real @ ( genClo1163638703lle_te @ Q @ I ) @ genClo1278781456e_beta ) @ T ) )
% 0.28/0.47 => ( ord_less_eq @ real @ ( genClo1163638703lle_te @ P @ I ) @ T ) ) ).
% 0.28/0.47
% 0.28/0.47 % rts2a
% 0.28/0.47 thf(fact_78_add__right__imp__eq,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( cancel_semigroup_add @ A )
% 0.28/0.47 => ! [B: A,A2: A,C: A] :
% 0.28/0.47 ( ( ( plus_plus @ A @ B @ A2 )
% 0.28/0.47 = ( plus_plus @ A @ C @ A2 ) )
% 0.28/0.47 => ( B = C ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % add_right_imp_eq
% 0.28/0.47 thf(fact_79_add__left__imp__eq,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( cancel_semigroup_add @ A )
% 0.28/0.47 => ! [A2: A,B: A,C: A] :
% 0.28/0.47 ( ( ( plus_plus @ A @ A2 @ B )
% 0.28/0.47 = ( plus_plus @ A @ A2 @ C ) )
% 0.28/0.47 => ( B = C ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % add_left_imp_eq
% 0.28/0.47 thf(fact_80_add_Oleft__commute,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ab_semigroup_add @ A )
% 0.28/0.47 => ! [B: A,A2: A,C: A] :
% 0.28/0.47 ( ( plus_plus @ A @ B @ ( plus_plus @ A @ A2 @ C ) )
% 0.28/0.47 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % add.left_commute
% 0.28/0.47 thf(fact_81_add_Ocommute,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ab_semigroup_add @ A )
% 0.28/0.47 => ( ( plus_plus @ A )
% 0.28/0.47 = ( ^ [A3: A,B3: A] : ( plus_plus @ A @ B3 @ A3 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % add.commute
% 0.28/0.47 thf(fact_82_add_Oright__cancel,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( group_add @ A )
% 0.28/0.47 => ! [B: A,A2: A,C: A] :
% 0.28/0.47 ( ( ( plus_plus @ A @ B @ A2 )
% 0.28/0.47 = ( plus_plus @ A @ C @ A2 ) )
% 0.28/0.47 = ( B = C ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % add.right_cancel
% 0.28/0.47 thf(fact_83_add_Oleft__cancel,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( group_add @ A )
% 0.28/0.47 => ! [A2: A,B: A,C: A] :
% 0.28/0.47 ( ( ( plus_plus @ A @ A2 @ B )
% 0.28/0.47 = ( plus_plus @ A @ A2 @ C ) )
% 0.28/0.47 = ( B = C ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % add.left_cancel
% 0.28/0.47 thf(fact_84_add_Oassoc,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( semigroup_add @ A )
% 0.28/0.47 => ! [A2: A,B: A,C: A] :
% 0.28/0.47 ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B ) @ C )
% 0.28/0.47 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % add.assoc
% 0.28/0.47 thf(fact_85_group__cancel_Oadd2,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( comm_monoid_add @ A )
% 0.28/0.47 => ! [B4: A,K: A,B: A,A2: A] :
% 0.28/0.47 ( ( B4
% 0.28/0.47 = ( plus_plus @ A @ K @ B ) )
% 0.28/0.47 => ( ( plus_plus @ A @ A2 @ B4 )
% 0.28/0.47 = ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % group_cancel.add2
% 0.28/0.47 thf(fact_86_group__cancel_Oadd1,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( comm_monoid_add @ A )
% 0.28/0.47 => ! [A4: A,K: A,A2: A,B: A] :
% 0.28/0.47 ( ( A4
% 0.28/0.47 = ( plus_plus @ A @ K @ A2 ) )
% 0.28/0.47 => ( ( plus_plus @ A @ A4 @ B )
% 0.28/0.47 = ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % group_cancel.add1
% 0.28/0.47 thf(fact_87_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ordere779506340up_add @ A )
% 0.28/0.47 => ! [I2: A,J: A,K: A,L: A] :
% 0.28/0.47 ( ( ( I2 = J )
% 0.28/0.47 & ( K = L ) )
% 0.28/0.47 => ( ( plus_plus @ A @ I2 @ K )
% 0.28/0.47 = ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % add_mono_thms_linordered_semiring(4)
% 0.28/0.47 thf(fact_88_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ab_semigroup_add @ A )
% 0.28/0.47 => ! [A2: A,B: A,C: A] :
% 0.28/0.47 ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B ) @ C )
% 0.28/0.47 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % ab_semigroup_add_class.add_ac(1)
% 0.28/0.47 thf(fact_89_one__reorient,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( one @ A )
% 0.28/0.47 => ! [X: A] :
% 0.28/0.47 ( ( ( one_one @ A )
% 0.28/0.47 = X )
% 0.28/0.47 = ( X
% 0.28/0.47 = ( one_one @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % one_reorient
% 0.28/0.47 thf(fact_90_diff__right__commute,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( cancel146912293up_add @ A )
% 0.28/0.47 => ! [A2: A,C: A,B: A] :
% 0.28/0.47 ( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C ) @ B )
% 0.28/0.47 = ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B ) @ C ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % diff_right_commute
% 0.28/0.47 thf(fact_91_diff__eq__diff__eq,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( group_add @ A )
% 0.28/0.47 => ! [A2: A,B: A,C: A,D: A] :
% 0.28/0.47 ( ( ( minus_minus @ A @ A2 @ B )
% 0.28/0.47 = ( minus_minus @ A @ C @ D ) )
% 0.28/0.47 => ( ( A2 = B )
% 0.28/0.47 = ( C = D ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % diff_eq_diff_eq
% 0.28/0.47 thf(fact_92_beta__bound1,axiom,
% 0.28/0.47 ! [P2: nat,I2: nat,Q2: nat] :
% 0.28/0.47 ( ( genClo1015804716orrect @ P2 @ ( genClo1163638703lle_te @ P2 @ ( plus_plus @ nat @ I2 @ ( one_one @ nat ) ) ) )
% 0.28/0.47 => ( ( genClo1015804716orrect @ Q2 @ ( genClo1163638703lle_te @ P2 @ ( plus_plus @ nat @ I2 @ ( one_one @ nat ) ) ) )
% 0.28/0.47 => ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( minus_minus @ real @ ( genClo1163638703lle_te @ P2 @ ( plus_plus @ nat @ I2 @ ( one_one @ nat ) ) ) @ ( genClo1163638703lle_te @ Q2 @ I2 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % beta_bound1
% 0.28/0.47 thf(fact_93_correct__closed,axiom,
% 0.28/0.47 ! [P: nat,S: real,T: real] :
% 0.28/0.47 ( ( ( ord_less_eq @ real @ S @ T )
% 0.28/0.47 & ( genClo1015804716orrect @ P @ T ) )
% 0.28/0.47 => ( genClo1015804716orrect @ P @ S ) ) ).
% 0.28/0.47
% 0.28/0.47 % correct_closed
% 0.28/0.47 thf(fact_94_add__le__imp__le__right,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ordere236663937imp_le @ A )
% 0.28/0.47 => ! [A2: A,C: A,B: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) )
% 0.28/0.47 => ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % add_le_imp_le_right
% 0.28/0.47 thf(fact_95_add__le__imp__le__left,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ordere236663937imp_le @ A )
% 0.28/0.47 => ! [C: A,A2: A,B: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) )
% 0.28/0.47 => ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % add_le_imp_le_left
% 0.28/0.47 thf(fact_96_le__iff__add,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( canoni770627133id_add @ A )
% 0.28/0.47 => ( ( ord_less_eq @ A )
% 0.28/0.47 = ( ^ [A3: A,B3: A] :
% 0.28/0.47 ? [C2: A] :
% 0.28/0.47 ( B3
% 0.28/0.47 = ( plus_plus @ A @ A3 @ C2 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % le_iff_add
% 0.28/0.47 thf(fact_97_add__right__mono,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ordere779506340up_add @ A )
% 0.28/0.47 => ! [A2: A,B: A,C: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ A2 @ B )
% 0.28/0.47 => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % add_right_mono
% 0.28/0.47 thf(fact_98_less__eqE,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( canoni770627133id_add @ A )
% 0.28/0.47 => ! [A2: A,B: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ A2 @ B )
% 0.28/0.47 => ~ ! [C3: A] :
% 0.28/0.47 ( B
% 0.28/0.47 != ( plus_plus @ A @ A2 @ C3 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % less_eqE
% 0.28/0.47 thf(fact_99_add__left__mono,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ordere779506340up_add @ A )
% 0.28/0.47 => ! [A2: A,B: A,C: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ A2 @ B )
% 0.28/0.47 => ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % add_left_mono
% 0.28/0.47 thf(fact_100_add__mono,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ordere779506340up_add @ A )
% 0.28/0.47 => ! [A2: A,B: A,C: A,D: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ A2 @ B )
% 0.28/0.47 => ( ( ord_less_eq @ A @ C @ D )
% 0.28/0.47 => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ D ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % add_mono
% 0.28/0.47 thf(fact_101_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ordere779506340up_add @ A )
% 0.28/0.47 => ! [I2: A,J: A,K: A,L: A] :
% 0.28/0.47 ( ( ( ord_less_eq @ A @ I2 @ J )
% 0.28/0.47 & ( ord_less_eq @ A @ K @ L ) )
% 0.28/0.47 => ( ord_less_eq @ A @ ( plus_plus @ A @ I2 @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % add_mono_thms_linordered_semiring(1)
% 0.28/0.47 thf(fact_102_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ordere779506340up_add @ A )
% 0.28/0.47 => ! [I2: A,J: A,K: A,L: A] :
% 0.28/0.47 ( ( ( I2 = J )
% 0.28/0.47 & ( ord_less_eq @ A @ K @ L ) )
% 0.28/0.47 => ( ord_less_eq @ A @ ( plus_plus @ A @ I2 @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % add_mono_thms_linordered_semiring(2)
% 0.28/0.47 thf(fact_103_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ordere779506340up_add @ A )
% 0.28/0.47 => ! [I2: A,J: A,K: A,L: A] :
% 0.28/0.47 ( ( ( ord_less_eq @ A @ I2 @ J )
% 0.28/0.47 & ( K = L ) )
% 0.28/0.47 => ( ord_less_eq @ A @ ( plus_plus @ A @ I2 @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % add_mono_thms_linordered_semiring(3)
% 0.28/0.47 thf(fact_104_diff__eq__diff__less__eq,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ordered_ab_group_add @ A )
% 0.28/0.47 => ! [A2: A,B: A,C: A,D: A] :
% 0.28/0.47 ( ( ( minus_minus @ A @ A2 @ B )
% 0.28/0.47 = ( minus_minus @ A @ C @ D ) )
% 0.28/0.47 => ( ( ord_less_eq @ A @ A2 @ B )
% 0.28/0.47 = ( ord_less_eq @ A @ C @ D ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % diff_eq_diff_less_eq
% 0.28/0.47 thf(fact_105_diff__right__mono,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ordered_ab_group_add @ A )
% 0.28/0.47 => ! [A2: A,B: A,C: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ A2 @ B )
% 0.28/0.47 => ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C ) @ ( minus_minus @ A @ B @ C ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % diff_right_mono
% 0.28/0.47 thf(fact_106_diff__left__mono,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ordered_ab_group_add @ A )
% 0.28/0.47 => ! [B: A,A2: A,C: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ B @ A2 )
% 0.28/0.47 => ( ord_less_eq @ A @ ( minus_minus @ A @ C @ A2 ) @ ( minus_minus @ A @ C @ B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % diff_left_mono
% 0.28/0.47 thf(fact_107_diff__mono,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ordered_ab_group_add @ A )
% 0.28/0.47 => ! [A2: A,B: A,D: A,C: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ A2 @ B )
% 0.28/0.47 => ( ( ord_less_eq @ A @ D @ C )
% 0.28/0.47 => ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C ) @ ( minus_minus @ A @ B @ D ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % diff_mono
% 0.28/0.47 thf(fact_108_add__implies__diff,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( cancel1352612707id_add @ A )
% 0.28/0.47 => ! [C: A,B: A,A2: A] :
% 0.28/0.47 ( ( ( plus_plus @ A @ C @ B )
% 0.28/0.47 = A2 )
% 0.28/0.47 => ( C
% 0.28/0.47 = ( minus_minus @ A @ A2 @ B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % add_implies_diff
% 0.28/0.47 thf(fact_109_diff__diff__add,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( cancel146912293up_add @ A )
% 0.28/0.47 => ! [A2: A,B: A,C: A] :
% 0.28/0.47 ( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B ) @ C )
% 0.28/0.47 = ( minus_minus @ A @ A2 @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % diff_diff_add
% 0.28/0.47 thf(fact_110_diff__add__eq__diff__diff__swap,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( group_add @ A )
% 0.28/0.47 => ! [A2: A,B: A,C: A] :
% 0.28/0.47 ( ( minus_minus @ A @ A2 @ ( plus_plus @ A @ B @ C ) )
% 0.28/0.47 = ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C ) @ B ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % diff_add_eq_diff_diff_swap
% 0.28/0.47 thf(fact_111_diff__add__eq,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ab_group_add @ A )
% 0.28/0.47 => ! [A2: A,B: A,C: A] :
% 0.28/0.47 ( ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B ) @ C )
% 0.28/0.47 = ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C ) @ B ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % diff_add_eq
% 0.28/0.47 thf(fact_112_diff__diff__eq2,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( group_add @ A )
% 0.28/0.47 => ! [A2: A,B: A,C: A] :
% 0.28/0.47 ( ( minus_minus @ A @ A2 @ ( minus_minus @ A @ B @ C ) )
% 0.28/0.47 = ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C ) @ B ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % diff_diff_eq2
% 0.28/0.47 thf(fact_113_add__diff__eq,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( group_add @ A )
% 0.28/0.47 => ! [A2: A,B: A,C: A] :
% 0.28/0.47 ( ( plus_plus @ A @ A2 @ ( minus_minus @ A @ B @ C ) )
% 0.28/0.47 = ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B ) @ C ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % add_diff_eq
% 0.28/0.47 thf(fact_114_eq__diff__eq,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( group_add @ A )
% 0.28/0.47 => ! [A2: A,C: A,B: A] :
% 0.28/0.47 ( ( A2
% 0.28/0.47 = ( minus_minus @ A @ C @ B ) )
% 0.28/0.47 = ( ( plus_plus @ A @ A2 @ B )
% 0.28/0.47 = C ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % eq_diff_eq
% 0.28/0.47 thf(fact_115_diff__eq__eq,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( group_add @ A )
% 0.28/0.47 => ! [A2: A,B: A,C: A] :
% 0.28/0.47 ( ( ( minus_minus @ A @ A2 @ B )
% 0.28/0.47 = C )
% 0.28/0.47 = ( A2
% 0.28/0.47 = ( plus_plus @ A @ C @ B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % diff_eq_eq
% 0.28/0.47 thf(fact_116_group__cancel_Osub1,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ab_group_add @ A )
% 0.28/0.47 => ! [A4: A,K: A,A2: A,B: A] :
% 0.28/0.47 ( ( A4
% 0.28/0.47 = ( plus_plus @ A @ K @ A2 ) )
% 0.28/0.47 => ( ( minus_minus @ A @ A4 @ B )
% 0.28/0.47 = ( plus_plus @ A @ K @ ( minus_minus @ A @ A2 @ B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % group_cancel.sub1
% 0.28/0.47 thf(fact_117_abs__ge__self,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ordere142940540dd_abs @ A )
% 0.28/0.47 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ ( abs_abs @ A @ A2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % abs_ge_self
% 0.28/0.47 thf(fact_118_abs__le__D1,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ordere142940540dd_abs @ A )
% 0.28/0.47 => ! [A2: A,B: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ B )
% 0.28/0.47 => ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % abs_le_D1
% 0.28/0.47 thf(fact_119_abs__one,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_idom @ A )
% 0.28/0.47 => ( ( abs_abs @ A @ ( one_one @ A ) )
% 0.28/0.47 = ( one_one @ A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % abs_one
% 0.28/0.47 thf(fact_120_abs__minus__commute,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ordere142940540dd_abs @ A )
% 0.28/0.47 => ! [A2: A,B: A] :
% 0.28/0.47 ( ( abs_abs @ A @ ( minus_minus @ A @ A2 @ B ) )
% 0.28/0.47 = ( abs_abs @ A @ ( minus_minus @ A @ B @ A2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % abs_minus_commute
% 0.28/0.47 thf(fact_121_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ordere623563068d_diff @ A )
% 0.28/0.47 => ! [A2: A,B: A,C: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ A2 @ B )
% 0.28/0.47 => ( ( ord_less_eq @ A @ A2 @ B )
% 0.28/0.47 => ( ( ( minus_minus @ A @ B @ A2 )
% 0.28/0.47 = C )
% 0.28/0.47 = ( B
% 0.28/0.47 = ( plus_plus @ A @ C @ A2 ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
% 0.28/0.47 thf(fact_122_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ordere623563068d_diff @ A )
% 0.28/0.47 => ! [A2: A,B: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ A2 @ B )
% 0.28/0.47 => ( ( plus_plus @ A @ A2 @ ( minus_minus @ A @ B @ A2 ) )
% 0.28/0.47 = B ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % ordered_cancel_comm_monoid_diff_class.add_diff_inverse
% 0.28/0.47 thf(fact_123_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ordere623563068d_diff @ A )
% 0.28/0.47 => ! [A2: A,B: A,C: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ A2 @ B )
% 0.28/0.47 => ( ( minus_minus @ A @ C @ ( minus_minus @ A @ B @ A2 ) )
% 0.28/0.47 = ( minus_minus @ A @ ( plus_plus @ A @ C @ A2 ) @ B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % ordered_cancel_comm_monoid_diff_class.diff_diff_right
% 0.28/0.47 thf(fact_124_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ordere623563068d_diff @ A )
% 0.28/0.47 => ! [A2: A,B: A,C: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ A2 @ B )
% 0.28/0.47 => ( ( minus_minus @ A @ ( plus_plus @ A @ B @ C ) @ A2 )
% 0.28/0.47 = ( plus_plus @ A @ ( minus_minus @ A @ B @ A2 ) @ C ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
% 0.28/0.47 thf(fact_125_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ordere623563068d_diff @ A )
% 0.28/0.47 => ! [A2: A,B: A,C: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ A2 @ B )
% 0.28/0.47 => ( ( plus_plus @ A @ ( minus_minus @ A @ B @ A2 ) @ C )
% 0.28/0.47 = ( minus_minus @ A @ ( plus_plus @ A @ B @ C ) @ A2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
% 0.28/0.47 thf(fact_126_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ordere623563068d_diff @ A )
% 0.28/0.47 => ! [A2: A,B: A,C: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ A2 @ B )
% 0.28/0.47 => ( ( minus_minus @ A @ ( plus_plus @ A @ C @ B ) @ A2 )
% 0.28/0.47 = ( plus_plus @ A @ C @ ( minus_minus @ A @ B @ A2 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % ordered_cancel_comm_monoid_diff_class.diff_add_assoc
% 0.28/0.47 thf(fact_127_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ordere623563068d_diff @ A )
% 0.28/0.47 => ! [A2: A,B: A,C: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ A2 @ B )
% 0.28/0.47 => ( ( plus_plus @ A @ C @ ( minus_minus @ A @ B @ A2 ) )
% 0.28/0.47 = ( minus_minus @ A @ ( plus_plus @ A @ C @ B ) @ A2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % ordered_cancel_comm_monoid_diff_class.add_diff_assoc
% 0.28/0.47 thf(fact_128_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ordere623563068d_diff @ A )
% 0.28/0.47 => ! [A2: A,B: A,C: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ A2 @ B )
% 0.28/0.47 => ( ( ord_less_eq @ A @ C @ ( minus_minus @ A @ B @ A2 ) )
% 0.28/0.47 = ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A2 ) @ B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % ordered_cancel_comm_monoid_diff_class.le_diff_conv2
% 0.28/0.47 thf(fact_129_le__add__diff,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ordere623563068d_diff @ A )
% 0.28/0.47 => ! [A2: A,B: A,C: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ A2 @ B )
% 0.28/0.47 => ( ord_less_eq @ A @ C @ ( minus_minus @ A @ ( plus_plus @ A @ B @ C ) @ A2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % le_add_diff
% 0.28/0.47 thf(fact_130_add__le__add__imp__diff__le,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_semidom @ A )
% 0.28/0.47 => ! [I2: A,K: A,N: A,J: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ ( plus_plus @ A @ I2 @ K ) @ N )
% 0.28/0.47 => ( ( ord_less_eq @ A @ N @ ( plus_plus @ A @ J @ K ) )
% 0.28/0.47 => ( ( ord_less_eq @ A @ ( plus_plus @ A @ I2 @ K ) @ N )
% 0.28/0.47 => ( ( ord_less_eq @ A @ N @ ( plus_plus @ A @ J @ K ) )
% 0.28/0.47 => ( ord_less_eq @ A @ ( minus_minus @ A @ N @ K ) @ J ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % add_le_add_imp_diff_le
% 0.28/0.47 thf(fact_131_diff__add,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ordere623563068d_diff @ A )
% 0.28/0.47 => ! [A2: A,B: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ A2 @ B )
% 0.28/0.47 => ( ( plus_plus @ A @ ( minus_minus @ A @ B @ A2 ) @ A2 )
% 0.28/0.47 = B ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % diff_add
% 0.28/0.47 thf(fact_132_add__le__imp__le__diff,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_semidom @ A )
% 0.28/0.47 => ! [I2: A,K: A,N: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ ( plus_plus @ A @ I2 @ K ) @ N )
% 0.28/0.47 => ( ord_less_eq @ A @ I2 @ ( minus_minus @ A @ N @ K ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % add_le_imp_le_diff
% 0.28/0.47 thf(fact_133_le__diff__eq,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ordered_ab_group_add @ A )
% 0.28/0.47 => ! [A2: A,C: A,B: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ A2 @ ( minus_minus @ A @ C @ B ) )
% 0.28/0.47 = ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B ) @ C ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % le_diff_eq
% 0.28/0.47 thf(fact_134_diff__le__eq,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ordered_ab_group_add @ A )
% 0.28/0.47 => ! [A2: A,B: A,C: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ B ) @ C )
% 0.28/0.47 = ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ C @ B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % diff_le_eq
% 0.28/0.47 thf(fact_135_abs__triangle__ineq,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ordere142940540dd_abs @ A )
% 0.28/0.47 => ! [A2: A,B: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( plus_plus @ A @ A2 @ B ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % abs_triangle_ineq
% 0.28/0.47 thf(fact_136_abs__triangle__ineq2__sym,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ordere142940540dd_abs @ A )
% 0.28/0.47 => ! [A2: A,B: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ B @ A2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % abs_triangle_ineq2_sym
% 0.28/0.47 thf(fact_137_abs__triangle__ineq3,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ordere142940540dd_abs @ A )
% 0.28/0.47 => ! [A2: A,B: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B ) ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ A2 @ B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % abs_triangle_ineq3
% 0.28/0.47 thf(fact_138_abs__triangle__ineq2,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ordere142940540dd_abs @ A )
% 0.28/0.47 => ! [A2: A,B: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ A2 @ B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % abs_triangle_ineq2
% 0.28/0.47 thf(fact_139_abs__diff__triangle__ineq,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ordere142940540dd_abs @ A )
% 0.28/0.47 => ! [A2: A,B: A,C: A,D: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B ) @ ( plus_plus @ A @ C @ D ) ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ ( minus_minus @ A @ A2 @ C ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ B @ D ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % abs_diff_triangle_ineq
% 0.28/0.47 thf(fact_140_abs__triangle__ineq4,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ordere142940540dd_abs @ A )
% 0.28/0.47 => ! [A2: A,B: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ A2 @ B ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % abs_triangle_ineq4
% 0.28/0.47 thf(fact_141_abs__diff__le__iff,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_idom @ A )
% 0.28/0.47 => ! [X: A,A2: A,R: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X @ A2 ) ) @ R )
% 0.28/0.47 = ( ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ R ) @ X )
% 0.28/0.47 & ( ord_less_eq @ A @ X @ ( plus_plus @ A @ A2 @ R ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % abs_diff_le_iff
% 0.28/0.47 thf(fact_142_posX,axiom,
% 0.28/0.47 ord_less_eq @ real @ ( zero_zero @ real ) @ ( minus_minus @ real @ ( genClo1161277105lle_PC @ l @ ( genClo1163638703lle_te @ p @ ( plus_plus @ nat @ i @ ( one_one @ nat ) ) ) ) @ ( genClo1161277105lle_PC @ l @ ( genClo1163638703lle_te @ q @ ( plus_plus @ nat @ i @ ( one_one @ nat ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % posX
% 0.28/0.47 thf(fact_143_diff__numeral__special_I9_J,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( neg_numeral @ A )
% 0.28/0.47 => ( ( minus_minus @ A @ ( one_one @ A ) @ ( one_one @ A ) )
% 0.28/0.47 = ( zero_zero @ A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % diff_numeral_special(9)
% 0.28/0.47 thf(fact_144_beta__bound2,axiom,
% 0.28/0.47 ! [P2: nat,I2: nat,Q2: nat] :
% 0.28/0.47 ( ( genClo1015804716orrect @ P2 @ ( genClo1163638703lle_te @ P2 @ ( plus_plus @ nat @ I2 @ ( one_one @ nat ) ) ) )
% 0.28/0.47 => ( ( genClo1015804716orrect @ Q2 @ ( genClo1163638703lle_te @ Q2 @ I2 ) )
% 0.28/0.47 => ( ord_less_eq @ real @ ( minus_minus @ real @ ( genClo1163638703lle_te @ P2 @ ( plus_plus @ nat @ I2 @ ( one_one @ nat ) ) ) @ ( genClo1163638703lle_te @ Q2 @ I2 ) ) @ ( plus_plus @ real @ genClo1650508560e_rmax @ genClo1278781456e_beta ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % beta_bound2
% 0.28/0.47 thf(fact_145_rts1d,axiom,
% 0.28/0.47 ! [P2: nat,I2: nat] :
% 0.28/0.47 ( ( genClo1015804716orrect @ P2 @ ( genClo1163638703lle_te @ P2 @ ( plus_plus @ nat @ I2 @ ( one_one @ nat ) ) ) )
% 0.28/0.47 => ( ord_less_eq @ real @ genClo1651033342e_rmin @ ( minus_minus @ real @ ( genClo1163638703lle_te @ P2 @ ( plus_plus @ nat @ I2 @ ( one_one @ nat ) ) ) @ ( genClo1163638703lle_te @ P2 @ I2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % rts1d
% 0.28/0.47 thf(fact_146_rts1,axiom,
% 0.28/0.47 ! [P: nat,T: real,I: nat] :
% 0.28/0.47 ( ( ( genClo1015804716orrect @ P @ T )
% 0.28/0.47 & ( ord_less_eq @ real @ ( genClo1163638703lle_te @ P @ ( plus_plus @ nat @ I @ ( one_one @ nat ) ) ) @ T ) )
% 0.28/0.47 => ( ord_less_eq @ real @ genClo1651033342e_rmin @ ( minus_minus @ real @ T @ ( genClo1163638703lle_te @ P @ I ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % rts1
% 0.28/0.47 thf(fact_147_rts0d,axiom,
% 0.28/0.47 ! [P2: nat,I2: nat] :
% 0.28/0.47 ( ( genClo1015804716orrect @ P2 @ ( genClo1163638703lle_te @ P2 @ ( plus_plus @ nat @ I2 @ ( one_one @ nat ) ) ) )
% 0.28/0.47 => ( ord_less_eq @ real @ ( minus_minus @ real @ ( genClo1163638703lle_te @ P2 @ ( plus_plus @ nat @ I2 @ ( one_one @ nat ) ) ) @ ( genClo1163638703lle_te @ P2 @ I2 ) ) @ genClo1650508560e_rmax ) ) ).
% 0.28/0.47
% 0.28/0.47 % rts0d
% 0.28/0.47 thf(fact_148_rts0,axiom,
% 0.28/0.47 ! [P: nat,T: real,I: nat] :
% 0.28/0.47 ( ( ( genClo1015804716orrect @ P @ T )
% 0.28/0.47 & ( ord_less_eq @ real @ T @ ( genClo1163638703lle_te @ P @ ( plus_plus @ nat @ I @ ( one_one @ nat ) ) ) ) )
% 0.28/0.47 => ( ord_less_eq @ real @ ( minus_minus @ real @ T @ ( genClo1163638703lle_te @ P @ I ) ) @ genClo1650508560e_rmax ) ) ).
% 0.28/0.47
% 0.28/0.47 % rts0
% 0.28/0.47 thf(fact_149_okRead2__def,axiom,
% 0.28/0.47 ( genClo293725282kRead2
% 0.28/0.47 = ( ^ [F2: nat > real,G2: nat > real,X3: real,Ppred: nat > $o] :
% 0.28/0.47 ! [P3: nat] :
% 0.28/0.47 ( ( Ppred @ P3 )
% 0.28/0.47 => ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( F2 @ P3 ) @ ( G2 @ P3 ) ) ) @ X3 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % okRead2_def
% 0.28/0.47 thf(fact_150_sin__bound__lemma,axiom,
% 0.28/0.47 ! [X: real,Y: real,U: real,V: real] :
% 0.28/0.47 ( ( X = Y )
% 0.28/0.47 => ( ( ord_less_eq @ real @ ( abs_abs @ real @ U ) @ V )
% 0.28/0.47 => ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( plus_plus @ real @ X @ U ) @ Y ) ) @ V ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % sin_bound_lemma
% 0.28/0.47 thf(fact_151_Nat_Oadd__diff__assoc,axiom,
% 0.28/0.47 ! [K: nat,J: nat,I2: nat] :
% 0.28/0.47 ( ( ord_less_eq @ nat @ K @ J )
% 0.28/0.47 => ( ( plus_plus @ nat @ I2 @ ( minus_minus @ nat @ J @ K ) )
% 0.28/0.47 = ( minus_minus @ nat @ ( plus_plus @ nat @ I2 @ J ) @ K ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % Nat.add_diff_assoc
% 0.28/0.47 thf(fact_152_le0,axiom,
% 0.28/0.47 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 0.28/0.47
% 0.28/0.47 % le0
% 0.28/0.47 thf(fact_153_bot__nat__0_Oextremum,axiom,
% 0.28/0.47 ! [A2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A2 ) ).
% 0.28/0.47
% 0.28/0.47 % bot_nat_0.extremum
% 0.28/0.47 thf(fact_154_add__is__0,axiom,
% 0.28/0.47 ! [M: nat,N: nat] :
% 0.28/0.47 ( ( ( plus_plus @ nat @ M @ N )
% 0.28/0.47 = ( zero_zero @ nat ) )
% 0.28/0.47 = ( ( M
% 0.28/0.47 = ( zero_zero @ nat ) )
% 0.28/0.47 & ( N
% 0.28/0.47 = ( zero_zero @ nat ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % add_is_0
% 0.28/0.47 thf(fact_155_Nat_Oadd__0__right,axiom,
% 0.28/0.47 ! [M: nat] :
% 0.28/0.47 ( ( plus_plus @ nat @ M @ ( zero_zero @ nat ) )
% 0.28/0.47 = M ) ).
% 0.28/0.47
% 0.28/0.47 % Nat.add_0_right
% 0.28/0.47 thf(fact_156_diff__0__eq__0,axiom,
% 0.28/0.47 ! [N: nat] :
% 0.28/0.47 ( ( minus_minus @ nat @ ( zero_zero @ nat ) @ N )
% 0.28/0.47 = ( zero_zero @ nat ) ) ).
% 0.28/0.47
% 0.28/0.47 % diff_0_eq_0
% 0.28/0.47 thf(fact_157_diff__self__eq__0,axiom,
% 0.28/0.47 ! [M: nat] :
% 0.28/0.47 ( ( minus_minus @ nat @ M @ M )
% 0.28/0.47 = ( zero_zero @ nat ) ) ).
% 0.28/0.47
% 0.28/0.47 % diff_self_eq_0
% 0.28/0.47 thf(fact_158_nat__add__left__cancel__le,axiom,
% 0.28/0.47 ! [K: nat,M: nat,N: nat] :
% 0.28/0.47 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N ) )
% 0.28/0.47 = ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.28/0.47
% 0.28/0.47 % nat_add_left_cancel_le
% 0.28/0.47 thf(fact_159_diff__is__0__eq,axiom,
% 0.28/0.47 ! [M: nat,N: nat] :
% 0.28/0.47 ( ( ( minus_minus @ nat @ M @ N )
% 0.28/0.47 = ( zero_zero @ nat ) )
% 0.28/0.47 = ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.28/0.47
% 0.28/0.47 % diff_is_0_eq
% 0.28/0.47 thf(fact_160_diff__is__0__eq_H,axiom,
% 0.28/0.47 ! [M: nat,N: nat] :
% 0.28/0.47 ( ( ord_less_eq @ nat @ M @ N )
% 0.28/0.47 => ( ( minus_minus @ nat @ M @ N )
% 0.28/0.47 = ( zero_zero @ nat ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % diff_is_0_eq'
% 0.28/0.47 thf(fact_161_diff__diff__cancel,axiom,
% 0.28/0.47 ! [I2: nat,N: nat] :
% 0.28/0.47 ( ( ord_less_eq @ nat @ I2 @ N )
% 0.28/0.47 => ( ( minus_minus @ nat @ N @ ( minus_minus @ nat @ N @ I2 ) )
% 0.28/0.47 = I2 ) ) ).
% 0.28/0.47
% 0.28/0.47 % diff_diff_cancel
% 0.28/0.47 thf(fact_162_diff__diff__left,axiom,
% 0.28/0.47 ! [I2: nat,J: nat,K: nat] :
% 0.28/0.47 ( ( minus_minus @ nat @ ( minus_minus @ nat @ I2 @ J ) @ K )
% 0.28/0.47 = ( minus_minus @ nat @ I2 @ ( plus_plus @ nat @ J @ K ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % diff_diff_left
% 0.28/0.47 thf(fact_163_Nat_Odiff__diff__right,axiom,
% 0.28/0.47 ! [K: nat,J: nat,I2: nat] :
% 0.28/0.47 ( ( ord_less_eq @ nat @ K @ J )
% 0.28/0.47 => ( ( minus_minus @ nat @ I2 @ ( minus_minus @ nat @ J @ K ) )
% 0.28/0.47 = ( minus_minus @ nat @ ( plus_plus @ nat @ I2 @ K ) @ J ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % Nat.diff_diff_right
% 0.28/0.47 thf(fact_164_Nat_Oadd__diff__assoc2,axiom,
% 0.28/0.47 ! [K: nat,J: nat,I2: nat] :
% 0.28/0.47 ( ( ord_less_eq @ nat @ K @ J )
% 0.28/0.47 => ( ( plus_plus @ nat @ ( minus_minus @ nat @ J @ K ) @ I2 )
% 0.28/0.47 = ( minus_minus @ nat @ ( plus_plus @ nat @ J @ I2 ) @ K ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % Nat.add_diff_assoc2
% 0.28/0.47 thf(fact_165_less__eq__nat_Osimps_I1_J,axiom,
% 0.28/0.47 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 0.28/0.47
% 0.28/0.47 % less_eq_nat.simps(1)
% 0.28/0.47 thf(fact_166_le__0__eq,axiom,
% 0.28/0.47 ! [N: nat] :
% 0.28/0.47 ( ( ord_less_eq @ nat @ N @ ( zero_zero @ nat ) )
% 0.28/0.47 = ( N
% 0.28/0.47 = ( zero_zero @ nat ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % le_0_eq
% 0.28/0.47 thf(fact_167_le__refl,axiom,
% 0.28/0.47 ! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% 0.28/0.47
% 0.28/0.47 % le_refl
% 0.28/0.47 thf(fact_168_le__trans,axiom,
% 0.28/0.47 ! [I2: nat,J: nat,K: nat] :
% 0.28/0.47 ( ( ord_less_eq @ nat @ I2 @ J )
% 0.28/0.47 => ( ( ord_less_eq @ nat @ J @ K )
% 0.28/0.47 => ( ord_less_eq @ nat @ I2 @ K ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % le_trans
% 0.28/0.47 thf(fact_169_eq__imp__le,axiom,
% 0.28/0.47 ! [M: nat,N: nat] :
% 0.28/0.47 ( ( M = N )
% 0.28/0.47 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.28/0.47
% 0.28/0.47 % eq_imp_le
% 0.28/0.47 thf(fact_170_le__antisym,axiom,
% 0.28/0.47 ! [M: nat,N: nat] :
% 0.28/0.47 ( ( ord_less_eq @ nat @ M @ N )
% 0.28/0.47 => ( ( ord_less_eq @ nat @ N @ M )
% 0.28/0.47 => ( M = N ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % le_antisym
% 0.28/0.47 thf(fact_171_eq__diff__iff,axiom,
% 0.28/0.47 ! [K: nat,M: nat,N: nat] :
% 0.28/0.47 ( ( ord_less_eq @ nat @ K @ M )
% 0.28/0.47 => ( ( ord_less_eq @ nat @ K @ N )
% 0.28/0.47 => ( ( ( minus_minus @ nat @ M @ K )
% 0.28/0.47 = ( minus_minus @ nat @ N @ K ) )
% 0.28/0.47 = ( M = N ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % eq_diff_iff
% 0.28/0.47 thf(fact_172_le__diff__iff,axiom,
% 0.28/0.47 ! [K: nat,M: nat,N: nat] :
% 0.28/0.47 ( ( ord_less_eq @ nat @ K @ M )
% 0.28/0.47 => ( ( ord_less_eq @ nat @ K @ N )
% 0.28/0.47 => ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
% 0.28/0.47 = ( ord_less_eq @ nat @ M @ N ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % le_diff_iff
% 0.28/0.47 thf(fact_173_diff__commute,axiom,
% 0.28/0.47 ! [I2: nat,J: nat,K: nat] :
% 0.28/0.47 ( ( minus_minus @ nat @ ( minus_minus @ nat @ I2 @ J ) @ K )
% 0.28/0.47 = ( minus_minus @ nat @ ( minus_minus @ nat @ I2 @ K ) @ J ) ) ).
% 0.28/0.47
% 0.28/0.47 % diff_commute
% 0.28/0.47 thf(fact_174_Nat_Odiff__diff__eq,axiom,
% 0.28/0.47 ! [K: nat,M: nat,N: nat] :
% 0.28/0.47 ( ( ord_less_eq @ nat @ K @ M )
% 0.28/0.47 => ( ( ord_less_eq @ nat @ K @ N )
% 0.28/0.47 => ( ( minus_minus @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
% 0.28/0.47 = ( minus_minus @ nat @ M @ N ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % Nat.diff_diff_eq
% 0.28/0.47 thf(fact_175_diff__le__mono,axiom,
% 0.28/0.47 ! [M: nat,N: nat,L: nat] :
% 0.28/0.47 ( ( ord_less_eq @ nat @ M @ N )
% 0.28/0.47 => ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ L ) @ ( minus_minus @ nat @ N @ L ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % diff_le_mono
% 0.28/0.47 thf(fact_176_diff__le__self,axiom,
% 0.28/0.47 ! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ N ) @ M ) ).
% 0.28/0.47
% 0.28/0.47 % diff_le_self
% 0.28/0.47 thf(fact_177_le__diff__iff_H,axiom,
% 0.28/0.47 ! [A2: nat,C: nat,B: nat] :
% 0.28/0.47 ( ( ord_less_eq @ nat @ A2 @ C )
% 0.28/0.47 => ( ( ord_less_eq @ nat @ B @ C )
% 0.28/0.47 => ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ C @ A2 ) @ ( minus_minus @ nat @ C @ B ) )
% 0.28/0.47 = ( ord_less_eq @ nat @ B @ A2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % le_diff_iff'
% 0.28/0.47 thf(fact_178_diff__le__mono2,axiom,
% 0.28/0.47 ! [M: nat,N: nat,L: nat] :
% 0.28/0.47 ( ( ord_less_eq @ nat @ M @ N )
% 0.28/0.47 => ( ord_less_eq @ nat @ ( minus_minus @ nat @ L @ N ) @ ( minus_minus @ nat @ L @ M ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % diff_le_mono2
% 0.28/0.47 thf(fact_179_nat__le__linear,axiom,
% 0.28/0.47 ! [M: nat,N: nat] :
% 0.28/0.47 ( ( ord_less_eq @ nat @ M @ N )
% 0.28/0.47 | ( ord_less_eq @ nat @ N @ M ) ) ).
% 0.28/0.47
% 0.28/0.47 % nat_le_linear
% 0.28/0.47 thf(fact_180_diffs0__imp__equal,axiom,
% 0.28/0.47 ! [M: nat,N: nat] :
% 0.28/0.47 ( ( ( minus_minus @ nat @ M @ N )
% 0.28/0.47 = ( zero_zero @ nat ) )
% 0.28/0.47 => ( ( ( minus_minus @ nat @ N @ M )
% 0.28/0.47 = ( zero_zero @ nat ) )
% 0.28/0.47 => ( M = N ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % diffs0_imp_equal
% 0.28/0.47 thf(fact_181_minus__nat_Odiff__0,axiom,
% 0.28/0.47 ! [M: nat] :
% 0.28/0.47 ( ( minus_minus @ nat @ M @ ( zero_zero @ nat ) )
% 0.28/0.47 = M ) ).
% 0.28/0.47
% 0.28/0.47 % minus_nat.diff_0
% 0.28/0.47 thf(fact_182_Nat_Oex__has__greatest__nat,axiom,
% 0.28/0.47 ! [P4: nat > $o,K: nat,B: nat] :
% 0.28/0.47 ( ( P4 @ K )
% 0.28/0.47 => ( ! [Y3: nat] :
% 0.28/0.47 ( ( P4 @ Y3 )
% 0.28/0.47 => ( ord_less_eq @ nat @ Y3 @ B ) )
% 0.28/0.47 => ? [X2: nat] :
% 0.28/0.47 ( ( P4 @ X2 )
% 0.28/0.47 & ! [Y4: nat] :
% 0.28/0.47 ( ( P4 @ Y4 )
% 0.28/0.47 => ( ord_less_eq @ nat @ Y4 @ X2 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % Nat.ex_has_greatest_nat
% 0.28/0.47 thf(fact_183_bot__nat__0_Oextremum__unique,axiom,
% 0.28/0.47 ! [A2: nat] :
% 0.28/0.47 ( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
% 0.28/0.47 = ( A2
% 0.28/0.47 = ( zero_zero @ nat ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % bot_nat_0.extremum_unique
% 0.28/0.47 thf(fact_184_bot__nat__0_Oextremum__uniqueI,axiom,
% 0.28/0.47 ! [A2: nat] :
% 0.28/0.47 ( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
% 0.28/0.47 => ( A2
% 0.28/0.47 = ( zero_zero @ nat ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % bot_nat_0.extremum_uniqueI
% 0.28/0.47 thf(fact_185_plus__nat_Oadd__0,axiom,
% 0.28/0.47 ! [N: nat] :
% 0.28/0.47 ( ( plus_plus @ nat @ ( zero_zero @ nat ) @ N )
% 0.28/0.47 = N ) ).
% 0.28/0.47
% 0.28/0.47 % plus_nat.add_0
% 0.28/0.47 thf(fact_186_add__eq__self__zero,axiom,
% 0.28/0.47 ! [M: nat,N: nat] :
% 0.28/0.47 ( ( ( plus_plus @ nat @ M @ N )
% 0.28/0.47 = M )
% 0.28/0.47 => ( N
% 0.28/0.47 = ( zero_zero @ nat ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % add_eq_self_zero
% 0.28/0.47 thf(fact_187_diff__add__0,axiom,
% 0.28/0.47 ! [N: nat,M: nat] :
% 0.28/0.47 ( ( minus_minus @ nat @ N @ ( plus_plus @ nat @ N @ M ) )
% 0.28/0.47 = ( zero_zero @ nat ) ) ).
% 0.28/0.47
% 0.28/0.47 % diff_add_0
% 0.28/0.47 thf(fact_188_synch0,axiom,
% 0.28/0.47 ! [P: nat] :
% 0.28/0.47 ( ( genClo1163638703lle_te @ P @ ( zero_zero @ nat ) )
% 0.28/0.47 = ( zero_zero @ real ) ) ).
% 0.28/0.47
% 0.28/0.47 % synch0
% 0.28/0.47 thf(fact_189_is__num__normalize_I1_J,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( neg_numeral @ A )
% 0.28/0.47 => ! [A2: A,B: A,C: A] :
% 0.28/0.47 ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B ) @ C )
% 0.28/0.47 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % is_num_normalize(1)
% 0.28/0.47 thf(fact_190_add__leE,axiom,
% 0.28/0.47 ! [M: nat,K: nat,N: nat] :
% 0.28/0.47 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K ) @ N )
% 0.28/0.47 => ~ ( ( ord_less_eq @ nat @ M @ N )
% 0.28/0.47 => ~ ( ord_less_eq @ nat @ K @ N ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % add_leE
% 0.28/0.47 thf(fact_191_le__add1,axiom,
% 0.28/0.47 ! [N: nat,M: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ N @ M ) ) ).
% 0.28/0.47
% 0.28/0.47 % le_add1
% 0.28/0.47 thf(fact_192_le__add2,axiom,
% 0.28/0.47 ! [N: nat,M: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ M @ N ) ) ).
% 0.28/0.47
% 0.28/0.47 % le_add2
% 0.28/0.47 thf(fact_193_add__leD1,axiom,
% 0.28/0.47 ! [M: nat,K: nat,N: nat] :
% 0.28/0.47 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K ) @ N )
% 0.28/0.47 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.28/0.47
% 0.28/0.47 % add_leD1
% 0.28/0.47 thf(fact_194_add__leD2,axiom,
% 0.28/0.47 ! [M: nat,K: nat,N: nat] :
% 0.28/0.47 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K ) @ N )
% 0.28/0.47 => ( ord_less_eq @ nat @ K @ N ) ) ).
% 0.28/0.47
% 0.28/0.47 % add_leD2
% 0.28/0.47 thf(fact_195_le__Suc__ex,axiom,
% 0.28/0.47 ! [K: nat,L: nat] :
% 0.28/0.47 ( ( ord_less_eq @ nat @ K @ L )
% 0.28/0.47 => ? [N2: nat] :
% 0.28/0.47 ( L
% 0.28/0.47 = ( plus_plus @ nat @ K @ N2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % le_Suc_ex
% 0.28/0.47 thf(fact_196_add__le__mono,axiom,
% 0.28/0.47 ! [I2: nat,J: nat,K: nat,L: nat] :
% 0.28/0.47 ( ( ord_less_eq @ nat @ I2 @ J )
% 0.28/0.47 => ( ( ord_less_eq @ nat @ K @ L )
% 0.28/0.47 => ( ord_less_eq @ nat @ ( plus_plus @ nat @ I2 @ K ) @ ( plus_plus @ nat @ J @ L ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % add_le_mono
% 0.28/0.47 thf(fact_197_add__le__mono1,axiom,
% 0.28/0.47 ! [I2: nat,J: nat,K: nat] :
% 0.28/0.47 ( ( ord_less_eq @ nat @ I2 @ J )
% 0.28/0.47 => ( ord_less_eq @ nat @ ( plus_plus @ nat @ I2 @ K ) @ ( plus_plus @ nat @ J @ K ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % add_le_mono1
% 0.28/0.47 thf(fact_198_trans__le__add1,axiom,
% 0.28/0.47 ! [I2: nat,J: nat,M: nat] :
% 0.28/0.47 ( ( ord_less_eq @ nat @ I2 @ J )
% 0.28/0.47 => ( ord_less_eq @ nat @ I2 @ ( plus_plus @ nat @ J @ M ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % trans_le_add1
% 0.28/0.47 thf(fact_199_trans__le__add2,axiom,
% 0.28/0.47 ! [I2: nat,J: nat,M: nat] :
% 0.28/0.47 ( ( ord_less_eq @ nat @ I2 @ J )
% 0.28/0.47 => ( ord_less_eq @ nat @ I2 @ ( plus_plus @ nat @ M @ J ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % trans_le_add2
% 0.28/0.47 thf(fact_200_nat__le__iff__add,axiom,
% 0.28/0.47 ( ( ord_less_eq @ nat )
% 0.28/0.47 = ( ^ [M2: nat,N3: nat] :
% 0.28/0.47 ? [K2: nat] :
% 0.28/0.47 ( N3
% 0.28/0.47 = ( plus_plus @ nat @ M2 @ K2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % nat_le_iff_add
% 0.28/0.47 thf(fact_201_Nat_Odiff__cancel,axiom,
% 0.28/0.47 ! [K: nat,M: nat,N: nat] :
% 0.28/0.47 ( ( minus_minus @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N ) )
% 0.28/0.47 = ( minus_minus @ nat @ M @ N ) ) ).
% 0.28/0.47
% 0.28/0.47 % Nat.diff_cancel
% 0.28/0.47 thf(fact_202_diff__cancel2,axiom,
% 0.28/0.47 ! [M: nat,K: nat,N: nat] :
% 0.28/0.47 ( ( minus_minus @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N @ K ) )
% 0.28/0.47 = ( minus_minus @ nat @ M @ N ) ) ).
% 0.28/0.47
% 0.28/0.47 % diff_cancel2
% 0.28/0.47 thf(fact_203_diff__add__inverse,axiom,
% 0.28/0.47 ! [N: nat,M: nat] :
% 0.28/0.47 ( ( minus_minus @ nat @ ( plus_plus @ nat @ N @ M ) @ N )
% 0.28/0.47 = M ) ).
% 0.28/0.47
% 0.28/0.47 % diff_add_inverse
% 0.28/0.47 thf(fact_204_diff__add__inverse2,axiom,
% 0.28/0.47 ! [M: nat,N: nat] :
% 0.28/0.47 ( ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ N )
% 0.28/0.47 = M ) ).
% 0.28/0.47
% 0.28/0.47 % diff_add_inverse2
% 0.28/0.47 thf(fact_205_Nat_Ole__imp__diff__is__add,axiom,
% 0.28/0.47 ! [I2: nat,J: nat,K: nat] :
% 0.28/0.47 ( ( ord_less_eq @ nat @ I2 @ J )
% 0.28/0.47 => ( ( ( minus_minus @ nat @ J @ I2 )
% 0.28/0.47 = K )
% 0.28/0.47 = ( J
% 0.28/0.47 = ( plus_plus @ nat @ K @ I2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % Nat.le_imp_diff_is_add
% 0.28/0.47 thf(fact_206_Nat_Odiff__add__assoc2,axiom,
% 0.28/0.47 ! [K: nat,J: nat,I2: nat] :
% 0.28/0.47 ( ( ord_less_eq @ nat @ K @ J )
% 0.28/0.47 => ( ( minus_minus @ nat @ ( plus_plus @ nat @ J @ I2 ) @ K )
% 0.28/0.47 = ( plus_plus @ nat @ ( minus_minus @ nat @ J @ K ) @ I2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % Nat.diff_add_assoc2
% 0.28/0.47 thf(fact_207_Nat_Odiff__add__assoc,axiom,
% 0.28/0.47 ! [K: nat,J: nat,I2: nat] :
% 0.28/0.47 ( ( ord_less_eq @ nat @ K @ J )
% 0.28/0.47 => ( ( minus_minus @ nat @ ( plus_plus @ nat @ I2 @ J ) @ K )
% 0.28/0.47 = ( plus_plus @ nat @ I2 @ ( minus_minus @ nat @ J @ K ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % Nat.diff_add_assoc
% 0.28/0.47 thf(fact_208_Nat_Ole__diff__conv2,axiom,
% 0.28/0.47 ! [K: nat,J: nat,I2: nat] :
% 0.28/0.47 ( ( ord_less_eq @ nat @ K @ J )
% 0.28/0.47 => ( ( ord_less_eq @ nat @ I2 @ ( minus_minus @ nat @ J @ K ) )
% 0.28/0.47 = ( ord_less_eq @ nat @ ( plus_plus @ nat @ I2 @ K ) @ J ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % Nat.le_diff_conv2
% 0.28/0.47 thf(fact_209_le__diff__conv,axiom,
% 0.28/0.47 ! [J: nat,K: nat,I2: nat] :
% 0.28/0.47 ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ J @ K ) @ I2 )
% 0.28/0.47 = ( ord_less_eq @ nat @ J @ ( plus_plus @ nat @ I2 @ K ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % le_diff_conv
% 0.28/0.47 thf(fact_210_PC__monotone,axiom,
% 0.28/0.47 ! [P: nat,S: real,T: real] :
% 0.28/0.47 ( ( ( genClo1015804716orrect @ P @ T )
% 0.28/0.47 & ( ord_less_eq @ real @ S @ T ) )
% 0.28/0.47 => ( ord_less_eq @ real @ ( genClo1161277105lle_PC @ P @ S ) @ ( genClo1161277105lle_PC @ P @ T ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % PC_monotone
% 0.28/0.47 thf(fact_211_eq__diff__eq_H,axiom,
% 0.28/0.47 ! [X: real,Y: real,Z2: real] :
% 0.28/0.47 ( ( X
% 0.28/0.47 = ( minus_minus @ real @ Y @ Z2 ) )
% 0.28/0.47 = ( Y
% 0.28/0.47 = ( plus_plus @ real @ X @ Z2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % eq_diff_eq'
% 0.28/0.47 thf(fact_212_le__numeral__extra_I3_J,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linord1659791738miring @ A )
% 0.28/0.47 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % le_numeral_extra(3)
% 0.28/0.47 thf(fact_213_nonoverlap,axiom,
% 0.28/0.47 ord_less_eq @ real @ genClo1278781456e_beta @ genClo1651033342e_rmin ).
% 0.28/0.47
% 0.28/0.47 % nonoverlap
% 0.28/0.47 thf(fact_214_le__numeral__extra_I4_J,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linord1659791738miring @ A )
% 0.28/0.47 => ( ord_less_eq @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % le_numeral_extra(4)
% 0.28/0.47 thf(fact_215_init,axiom,
% 0.28/0.47 ! [P: nat] :
% 0.28/0.47 ( ( genClo1015804716orrect @ P @ ( zero_zero @ real ) )
% 0.28/0.47 => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( genClo1161277105lle_PC @ P @ ( zero_zero @ real ) ) )
% 0.28/0.47 & ( ord_less_eq @ real @ ( genClo1161277105lle_PC @ P @ ( zero_zero @ real ) ) @ genClo872980712lle_mu ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % init
% 0.28/0.47 thf(fact_216_bound1,axiom,
% 0.28/0.47 ord_less_eq @ real @ ( minus_minus @ real @ ( genClo1161277105lle_PC @ l @ ( genClo1163638703lle_te @ p @ ( plus_plus @ nat @ i @ ( one_one @ nat ) ) ) ) @ ( genClo1161277105lle_PC @ l @ ( genClo1163638703lle_te @ q @ ( plus_plus @ nat @ i @ ( one_one @ nat ) ) ) ) ) @ ( times_times @ real @ ( minus_minus @ real @ ( genClo1163638703lle_te @ p @ ( plus_plus @ nat @ i @ ( one_one @ nat ) ) ) @ ( genClo1163638703lle_te @ q @ ( plus_plus @ nat @ i @ ( one_one @ nat ) ) ) ) @ ( plus_plus @ real @ ( one_one @ real ) @ genClo1144207539le_rho ) ) ).
% 0.28/0.47
% 0.28/0.47 % bound1
% 0.28/0.47 thf(fact_217_minus__apply,axiom,
% 0.28/0.47 ! [B2: $tType,A: $tType] :
% 0.28/0.47 ( ( minus @ B2 )
% 0.28/0.47 => ( ( minus_minus @ ( A > B2 ) )
% 0.28/0.47 = ( ^ [A5: A > B2,B5: A > B2,X3: A] : ( minus_minus @ B2 @ ( A5 @ X3 ) @ ( B5 @ X3 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % minus_apply
% 0.28/0.47 thf(fact_218_mult__cancel__right,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( semiri1923998003cancel @ A )
% 0.28/0.47 => ! [A2: A,C: A,B: A] :
% 0.28/0.47 ( ( ( times_times @ A @ A2 @ C )
% 0.28/0.47 = ( times_times @ A @ B @ C ) )
% 0.28/0.47 = ( ( C
% 0.28/0.47 = ( zero_zero @ A ) )
% 0.28/0.47 | ( A2 = B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % mult_cancel_right
% 0.28/0.47 thf(fact_219_mult__cancel__left,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( semiri1923998003cancel @ A )
% 0.28/0.47 => ! [C: A,A2: A,B: A] :
% 0.28/0.47 ( ( ( times_times @ A @ C @ A2 )
% 0.28/0.47 = ( times_times @ A @ C @ B ) )
% 0.28/0.47 = ( ( C
% 0.28/0.47 = ( zero_zero @ A ) )
% 0.28/0.47 | ( A2 = B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % mult_cancel_left
% 0.28/0.47 thf(fact_220_mult__eq__0__iff,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( semiri1193490041visors @ A )
% 0.28/0.47 => ! [A2: A,B: A] :
% 0.28/0.47 ( ( ( times_times @ A @ A2 @ B )
% 0.28/0.47 = ( zero_zero @ A ) )
% 0.28/0.47 = ( ( A2
% 0.28/0.47 = ( zero_zero @ A ) )
% 0.28/0.47 | ( B
% 0.28/0.47 = ( zero_zero @ A ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % mult_eq_0_iff
% 0.28/0.47 thf(fact_221_mult__zero__right,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( mult_zero @ A )
% 0.28/0.47 => ! [A2: A] :
% 0.28/0.47 ( ( times_times @ A @ A2 @ ( zero_zero @ A ) )
% 0.28/0.47 = ( zero_zero @ A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % mult_zero_right
% 0.28/0.47 thf(fact_222_mult__zero__left,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( mult_zero @ A )
% 0.28/0.47 => ! [A2: A] :
% 0.28/0.47 ( ( times_times @ A @ ( zero_zero @ A ) @ A2 )
% 0.28/0.47 = ( zero_zero @ A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % mult_zero_left
% 0.28/0.47 thf(fact_223_mult_Oleft__neutral,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( monoid_mult @ A )
% 0.28/0.47 => ! [A2: A] :
% 0.28/0.47 ( ( times_times @ A @ ( one_one @ A ) @ A2 )
% 0.28/0.47 = A2 ) ) ).
% 0.28/0.47
% 0.28/0.47 % mult.left_neutral
% 0.28/0.47 thf(fact_224_mult_Oright__neutral,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( monoid_mult @ A )
% 0.28/0.47 => ! [A2: A] :
% 0.28/0.47 ( ( times_times @ A @ A2 @ ( one_one @ A ) )
% 0.28/0.47 = A2 ) ) ).
% 0.28/0.47
% 0.28/0.47 % mult.right_neutral
% 0.28/0.47 thf(fact_225_abs__mult__self__eq,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_idom @ A )
% 0.28/0.47 => ! [A2: A] :
% 0.28/0.47 ( ( times_times @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ A2 ) )
% 0.28/0.47 = ( times_times @ A @ A2 @ A2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % abs_mult_self_eq
% 0.28/0.47 thf(fact_226_mult__cancel__left1,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ring_11004092258visors @ A )
% 0.28/0.47 => ! [C: A,B: A] :
% 0.28/0.47 ( ( C
% 0.28/0.47 = ( times_times @ A @ C @ B ) )
% 0.28/0.47 = ( ( C
% 0.28/0.47 = ( zero_zero @ A ) )
% 0.28/0.47 | ( B
% 0.28/0.47 = ( one_one @ A ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % mult_cancel_left1
% 0.28/0.47 thf(fact_227_mult__cancel__left2,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ring_11004092258visors @ A )
% 0.28/0.47 => ! [C: A,A2: A] :
% 0.28/0.47 ( ( ( times_times @ A @ C @ A2 )
% 0.28/0.47 = C )
% 0.28/0.47 = ( ( C
% 0.28/0.47 = ( zero_zero @ A ) )
% 0.28/0.47 | ( A2
% 0.28/0.47 = ( one_one @ A ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % mult_cancel_left2
% 0.28/0.47 thf(fact_228_mult__cancel__right1,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ring_11004092258visors @ A )
% 0.28/0.47 => ! [C: A,B: A] :
% 0.28/0.47 ( ( C
% 0.28/0.47 = ( times_times @ A @ B @ C ) )
% 0.28/0.47 = ( ( C
% 0.28/0.47 = ( zero_zero @ A ) )
% 0.28/0.47 | ( B
% 0.28/0.47 = ( one_one @ A ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % mult_cancel_right1
% 0.28/0.47 thf(fact_229_mult__cancel__right2,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ring_11004092258visors @ A )
% 0.28/0.47 => ! [A2: A,C: A] :
% 0.28/0.47 ( ( ( times_times @ A @ A2 @ C )
% 0.28/0.47 = C )
% 0.28/0.47 = ( ( C
% 0.28/0.47 = ( zero_zero @ A ) )
% 0.28/0.47 | ( A2
% 0.28/0.47 = ( one_one @ A ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % mult_cancel_right2
% 0.28/0.47 thf(fact_230_mult__right__cancel,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( semiri1923998003cancel @ A )
% 0.28/0.47 => ! [C: A,A2: A,B: A] :
% 0.28/0.47 ( ( C
% 0.28/0.47 != ( zero_zero @ A ) )
% 0.28/0.47 => ( ( ( times_times @ A @ A2 @ C )
% 0.28/0.47 = ( times_times @ A @ B @ C ) )
% 0.28/0.47 = ( A2 = B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % mult_right_cancel
% 0.28/0.47 thf(fact_231_mult__left__cancel,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( semiri1923998003cancel @ A )
% 0.28/0.47 => ! [C: A,A2: A,B: A] :
% 0.28/0.47 ( ( C
% 0.28/0.47 != ( zero_zero @ A ) )
% 0.28/0.47 => ( ( ( times_times @ A @ C @ A2 )
% 0.28/0.47 = ( times_times @ A @ C @ B ) )
% 0.28/0.47 = ( A2 = B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % mult_left_cancel
% 0.28/0.47 thf(fact_232_no__zero__divisors,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( semiri1193490041visors @ A )
% 0.28/0.47 => ! [A2: A,B: A] :
% 0.28/0.47 ( ( A2
% 0.28/0.47 != ( zero_zero @ A ) )
% 0.28/0.47 => ( ( B
% 0.28/0.47 != ( zero_zero @ A ) )
% 0.28/0.47 => ( ( times_times @ A @ A2 @ B )
% 0.28/0.47 != ( zero_zero @ A ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % no_zero_divisors
% 0.28/0.47 thf(fact_233_divisors__zero,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( semiri1193490041visors @ A )
% 0.28/0.47 => ! [A2: A,B: A] :
% 0.28/0.47 ( ( ( times_times @ A @ A2 @ B )
% 0.28/0.47 = ( zero_zero @ A ) )
% 0.28/0.47 => ( ( A2
% 0.28/0.47 = ( zero_zero @ A ) )
% 0.28/0.47 | ( B
% 0.28/0.47 = ( zero_zero @ A ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % divisors_zero
% 0.28/0.47 thf(fact_234_mult__not__zero,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( mult_zero @ A )
% 0.28/0.47 => ! [A2: A,B: A] :
% 0.28/0.47 ( ( ( times_times @ A @ A2 @ B )
% 0.28/0.47 != ( zero_zero @ A ) )
% 0.28/0.47 => ( ( A2
% 0.28/0.47 != ( zero_zero @ A ) )
% 0.28/0.47 & ( B
% 0.28/0.47 != ( zero_zero @ A ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % mult_not_zero
% 0.28/0.47 thf(fact_235_ring__class_Oring__distribs_I2_J,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ring @ A )
% 0.28/0.47 => ! [A2: A,B: A,C: A] :
% 0.28/0.47 ( ( times_times @ A @ ( plus_plus @ A @ A2 @ B ) @ C )
% 0.28/0.47 = ( plus_plus @ A @ ( times_times @ A @ A2 @ C ) @ ( times_times @ A @ B @ C ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % ring_class.ring_distribs(2)
% 0.28/0.47 thf(fact_236_ring__class_Oring__distribs_I1_J,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ring @ A )
% 0.28/0.47 => ! [A2: A,B: A,C: A] :
% 0.28/0.47 ( ( times_times @ A @ A2 @ ( plus_plus @ A @ B @ C ) )
% 0.28/0.47 = ( plus_plus @ A @ ( times_times @ A @ A2 @ B ) @ ( times_times @ A @ A2 @ C ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % ring_class.ring_distribs(1)
% 0.28/0.47 thf(fact_237_comm__semiring__class_Odistrib,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( comm_semiring @ A )
% 0.28/0.47 => ! [A2: A,B: A,C: A] :
% 0.28/0.47 ( ( times_times @ A @ ( plus_plus @ A @ A2 @ B ) @ C )
% 0.28/0.47 = ( plus_plus @ A @ ( times_times @ A @ A2 @ C ) @ ( times_times @ A @ B @ C ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % comm_semiring_class.distrib
% 0.28/0.47 thf(fact_238_distrib__left,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( semiring @ A )
% 0.28/0.47 => ! [A2: A,B: A,C: A] :
% 0.28/0.47 ( ( times_times @ A @ A2 @ ( plus_plus @ A @ B @ C ) )
% 0.28/0.47 = ( plus_plus @ A @ ( times_times @ A @ A2 @ B ) @ ( times_times @ A @ A2 @ C ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % distrib_left
% 0.28/0.47 thf(fact_239_distrib__right,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( semiring @ A )
% 0.28/0.47 => ! [A2: A,B: A,C: A] :
% 0.28/0.47 ( ( times_times @ A @ ( plus_plus @ A @ A2 @ B ) @ C )
% 0.28/0.47 = ( plus_plus @ A @ ( times_times @ A @ A2 @ C ) @ ( times_times @ A @ B @ C ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % distrib_right
% 0.28/0.47 thf(fact_240_combine__common__factor,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( semiring @ A )
% 0.28/0.47 => ! [A2: A,E: A,B: A,C: A] :
% 0.28/0.47 ( ( plus_plus @ A @ ( times_times @ A @ A2 @ E ) @ ( plus_plus @ A @ ( times_times @ A @ B @ E ) @ C ) )
% 0.28/0.47 = ( plus_plus @ A @ ( times_times @ A @ ( plus_plus @ A @ A2 @ B ) @ E ) @ C ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % combine_common_factor
% 0.28/0.47 thf(fact_241_comm__monoid__mult__class_Omult__1,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( comm_monoid_mult @ A )
% 0.28/0.47 => ! [A2: A] :
% 0.28/0.47 ( ( times_times @ A @ ( one_one @ A ) @ A2 )
% 0.28/0.47 = A2 ) ) ).
% 0.28/0.47
% 0.28/0.47 % comm_monoid_mult_class.mult_1
% 0.28/0.47 thf(fact_242_mult_Ocomm__neutral,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( comm_monoid_mult @ A )
% 0.28/0.47 => ! [A2: A] :
% 0.28/0.47 ( ( times_times @ A @ A2 @ ( one_one @ A ) )
% 0.28/0.47 = A2 ) ) ).
% 0.28/0.47
% 0.28/0.47 % mult.comm_neutral
% 0.28/0.47 thf(fact_243_left__diff__distrib,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ring @ A )
% 0.28/0.47 => ! [A2: A,B: A,C: A] :
% 0.28/0.47 ( ( times_times @ A @ ( minus_minus @ A @ A2 @ B ) @ C )
% 0.28/0.47 = ( minus_minus @ A @ ( times_times @ A @ A2 @ C ) @ ( times_times @ A @ B @ C ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % left_diff_distrib
% 0.28/0.47 thf(fact_244_right__diff__distrib,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ring @ A )
% 0.28/0.47 => ! [A2: A,B: A,C: A] :
% 0.28/0.47 ( ( times_times @ A @ A2 @ ( minus_minus @ A @ B @ C ) )
% 0.28/0.47 = ( minus_minus @ A @ ( times_times @ A @ A2 @ B ) @ ( times_times @ A @ A2 @ C ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % right_diff_distrib
% 0.28/0.47 thf(fact_245_left__diff__distrib_H,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( comm_s1003936772cancel @ A )
% 0.28/0.47 => ! [B: A,C: A,A2: A] :
% 0.28/0.47 ( ( times_times @ A @ ( minus_minus @ A @ B @ C ) @ A2 )
% 0.28/0.47 = ( minus_minus @ A @ ( times_times @ A @ B @ A2 ) @ ( times_times @ A @ C @ A2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % left_diff_distrib'
% 0.28/0.47 thf(fact_246_right__diff__distrib_H,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( comm_s1003936772cancel @ A )
% 0.28/0.47 => ! [A2: A,B: A,C: A] :
% 0.28/0.47 ( ( times_times @ A @ A2 @ ( minus_minus @ A @ B @ C ) )
% 0.28/0.47 = ( minus_minus @ A @ ( times_times @ A @ A2 @ B ) @ ( times_times @ A @ A2 @ C ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % right_diff_distrib'
% 0.28/0.47 thf(fact_247_abs__mult,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( idom_abs_sgn @ A )
% 0.28/0.47 => ! [A2: A,B: A] :
% 0.28/0.47 ( ( abs_abs @ A @ ( times_times @ A @ A2 @ B ) )
% 0.28/0.47 = ( times_times @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % abs_mult
% 0.28/0.47 thf(fact_248_mult_Oleft__commute,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ab_semigroup_mult @ A )
% 0.28/0.47 => ! [B: A,A2: A,C: A] :
% 0.28/0.47 ( ( times_times @ A @ B @ ( times_times @ A @ A2 @ C ) )
% 0.28/0.47 = ( times_times @ A @ A2 @ ( times_times @ A @ B @ C ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % mult.left_commute
% 0.28/0.47 thf(fact_249_mult_Ocommute,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ab_semigroup_mult @ A )
% 0.28/0.47 => ( ( times_times @ A )
% 0.28/0.47 = ( ^ [A3: A,B3: A] : ( times_times @ A @ B3 @ A3 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % mult.commute
% 0.28/0.47 thf(fact_250_mult_Oassoc,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( semigroup_mult @ A )
% 0.28/0.47 => ! [A2: A,B: A,C: A] :
% 0.28/0.47 ( ( times_times @ A @ ( times_times @ A @ A2 @ B ) @ C )
% 0.28/0.47 = ( times_times @ A @ A2 @ ( times_times @ A @ B @ C ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % mult.assoc
% 0.28/0.47 thf(fact_251_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ab_semigroup_mult @ A )
% 0.28/0.47 => ! [A2: A,B: A,C: A] :
% 0.28/0.47 ( ( times_times @ A @ ( times_times @ A @ A2 @ B ) @ C )
% 0.28/0.47 = ( times_times @ A @ A2 @ ( times_times @ A @ B @ C ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % ab_semigroup_mult_class.mult_ac(1)
% 0.28/0.47 thf(fact_252_mult__mono,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ordered_semiring @ A )
% 0.28/0.47 => ! [A2: A,B: A,C: A,D: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ A2 @ B )
% 0.28/0.47 => ( ( ord_less_eq @ A @ C @ D )
% 0.28/0.47 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B )
% 0.28/0.47 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C )
% 0.28/0.47 => ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C ) @ ( times_times @ A @ B @ D ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % mult_mono
% 0.28/0.47
% 0.28/0.47 % Type constructors (71)
% 0.28/0.47 thf(tcon_fun___Groups_Ominus,axiom,
% 0.28/0.47 ! [A6: $tType,A7: $tType] :
% 0.28/0.47 ( ( minus @ A7 )
% 0.28/0.47 => ( minus @ ( A6 > A7 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
% 0.28/0.47 ordere516151231imp_le @ nat ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel,axiom,
% 0.28/0.47 semiri1923998003cancel @ nat ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__diff,axiom,
% 0.28/0.47 ordere623563068d_diff @ nat ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
% 0.28/0.47 ordere236663937imp_le @ nat ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 0.28/0.47 canoni770627133id_add @ nat ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring,axiom,
% 0.28/0.47 linord1659791738miring @ nat ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors,axiom,
% 0.28/0.47 semiri1193490041visors @ nat ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add,axiom,
% 0.28/0.47 ordere779506340up_add @ nat ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add,axiom,
% 0.28/0.47 ordere216010020id_add @ nat ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add,axiom,
% 0.28/0.47 cancel146912293up_add @ nat ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add,axiom,
% 0.28/0.47 cancel1352612707id_add @ nat ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1__cancel,axiom,
% 0.28/0.47 comm_s1003936772cancel @ nat ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add,axiom,
% 0.28/0.47 cancel_semigroup_add @ nat ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Nat_Onat___Rings_Olinordered__semidom,axiom,
% 0.28/0.47 linordered_semidom @ nat ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Nat_Onat___Groups_Oab__semigroup__mult,axiom,
% 0.28/0.47 ab_semigroup_mult @ nat ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__mult,axiom,
% 0.28/0.47 comm_monoid_mult @ nat ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__diff,axiom,
% 0.28/0.47 comm_monoid_diff @ nat ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Nat_Onat___Groups_Oab__semigroup__add,axiom,
% 0.28/0.47 ab_semigroup_add @ nat ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Nat_Onat___Rings_Oordered__semiring,axiom,
% 0.28/0.47 ordered_semiring @ nat ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add,axiom,
% 0.28/0.47 comm_monoid_add @ nat ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Nat_Onat___Groups_Osemigroup__mult,axiom,
% 0.28/0.47 semigroup_mult @ nat ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Nat_Onat___Groups_Osemigroup__add,axiom,
% 0.28/0.47 semigroup_add @ nat ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Nat_Onat___Rings_Ocomm__semiring,axiom,
% 0.28/0.47 comm_semiring @ nat ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Nat_Onat___Rings_Ozero__neq__one,axiom,
% 0.28/0.47 zero_neq_one @ nat ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Nat_Onat___Groups_Omonoid__mult,axiom,
% 0.28/0.47 monoid_mult @ nat ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Nat_Onat___Groups_Omonoid__add,axiom,
% 0.28/0.47 monoid_add @ nat ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Nat_Onat___Rings_Omult__zero,axiom,
% 0.28/0.47 mult_zero @ nat ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Nat_Onat___Rings_Osemiring,axiom,
% 0.28/0.47 semiring @ nat ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Nat_Onat___Groups_Ominus_1,axiom,
% 0.28/0.47 minus @ nat ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Nat_Onat___Groups_Ozero,axiom,
% 0.28/0.47 zero @ nat ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Nat_Onat___Groups_Oone,axiom,
% 0.28/0.47 one @ nat ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_HOL_Obool___Groups_Ominus_2,axiom,
% 0.28/0.47 minus @ $o ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__monoid__add__imp__le_3,axiom,
% 0.28/0.47 ordere516151231imp_le @ real ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors__cancel_4,axiom,
% 0.28/0.47 semiri1923998003cancel @ real ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add__imp__le_5,axiom,
% 0.28/0.47 ordere236663937imp_le @ real ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_6,axiom,
% 0.28/0.47 linord1659791738miring @ real ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors_7,axiom,
% 0.28/0.47 semiri1193490041visors @ real ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add_8,axiom,
% 0.28/0.47 ordere779506340up_add @ real ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add__abs,axiom,
% 0.28/0.47 ordere142940540dd_abs @ real ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Real_Oreal___Groups_Oordered__comm__monoid__add_9,axiom,
% 0.28/0.47 ordere216010020id_add @ real ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Real_Oreal___Groups_Olinordered__ab__group__add,axiom,
% 0.28/0.47 linord219039673up_add @ real ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Real_Oreal___Groups_Ocancel__ab__semigroup__add_10,axiom,
% 0.28/0.47 cancel146912293up_add @ real ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Real_Oreal___Rings_Oring__1__no__zero__divisors,axiom,
% 0.28/0.47 ring_11004092258visors @ real ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Real_Oreal___Groups_Ocancel__comm__monoid__add_11,axiom,
% 0.28/0.47 cancel1352612707id_add @ real ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Real_Oreal___Rings_Ocomm__semiring__1__cancel_12,axiom,
% 0.28/0.47 comm_s1003936772cancel @ real ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add,axiom,
% 0.28/0.47 ordered_ab_group_add @ real ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Real_Oreal___Groups_Ocancel__semigroup__add_13,axiom,
% 0.28/0.47 cancel_semigroup_add @ real ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Real_Oreal___Rings_Olinordered__semidom_14,axiom,
% 0.28/0.47 linordered_semidom @ real ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Real_Oreal___Groups_Oab__semigroup__mult_15,axiom,
% 0.28/0.47 ab_semigroup_mult @ real ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Real_Oreal___Groups_Ocomm__monoid__mult_16,axiom,
% 0.28/0.47 comm_monoid_mult @ real ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Real_Oreal___Groups_Oab__semigroup__add_17,axiom,
% 0.28/0.47 ab_semigroup_add @ real ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Real_Oreal___Rings_Oordered__semiring_18,axiom,
% 0.28/0.47 ordered_semiring @ real ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Real_Oreal___Groups_Ocomm__monoid__add_19,axiom,
% 0.28/0.47 comm_monoid_add @ real ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Real_Oreal___Rings_Olinordered__idom,axiom,
% 0.28/0.47 linordered_idom @ real ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Real_Oreal___Groups_Osemigroup__mult_20,axiom,
% 0.28/0.47 semigroup_mult @ real ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Real_Oreal___Groups_Osemigroup__add_21,axiom,
% 0.28/0.47 semigroup_add @ real ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Real_Oreal___Rings_Ocomm__semiring_22,axiom,
% 0.28/0.47 comm_semiring @ real ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Real_Oreal___Groups_Oab__group__add,axiom,
% 0.28/0.47 ab_group_add @ real ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Real_Oreal___Rings_Ozero__neq__one_23,axiom,
% 0.28/0.47 zero_neq_one @ real ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Real_Oreal___Rings_Oidom__abs__sgn,axiom,
% 0.28/0.47 idom_abs_sgn @ real ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Real_Oreal___Groups_Omonoid__mult_24,axiom,
% 0.28/0.47 monoid_mult @ real ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Real_Oreal___Groups_Omonoid__add_25,axiom,
% 0.28/0.47 monoid_add @ real ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Real_Oreal___Groups_Ogroup__add,axiom,
% 0.28/0.47 group_add @ real ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Real_Oreal___Rings_Omult__zero_26,axiom,
% 0.28/0.47 mult_zero @ real ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Real_Oreal___Num_Oneg__numeral,axiom,
% 0.28/0.47 neg_numeral @ real ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Real_Oreal___Rings_Osemiring_27,axiom,
% 0.28/0.47 semiring @ real ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Real_Oreal___Groups_Ominus_28,axiom,
% 0.28/0.47 minus @ real ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Real_Oreal___Groups_Ozero_29,axiom,
% 0.28/0.47 zero @ real ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Real_Oreal___Rings_Oring,axiom,
% 0.28/0.47 ring @ real ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Real_Oreal___Groups_Oone_30,axiom,
% 0.28/0.47 one @ real ).
% 0.28/0.47
% 0.28/0.47 % Conjectures (1)
% 0.28/0.47 thf(conj_0,conjecture,
% 0.28/0.47 ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( genClo1163638703lle_te @ p @ ( plus_plus @ nat @ i @ ( one_one @ nat ) ) ) @ ( genClo1163638703lle_te @ q @ ( plus_plus @ nat @ i @ ( one_one @ nat ) ) ) ) ) @ genClo1278781456e_beta ).
% 0.28/0.47
% 0.28/0.47 %------------------------------------------------------------------------------
% 0.28/0.47 ------------------------
% 2.57/1.56 ---- Embedded in THF ---
% 2.57/1.56 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.57/1.56 %%% Generated on Tue Feb 24 08:25:34 EST 2026
% 2.57/1.56 %%% using '$$dhol' embedding, version 1.3.0.
% 2.57/1.56 %%% Logic specification used:
% 2.57/1.56 %%% thf(spec, logic, $$dhol).
% 2.57/1.56
% 2.57/1.56 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.XXNzzqTI4f/DTF2DTF9062.p
% See solution above
% 2.57/1.56 ------------------------
% 2.72/1.60 ---- Cleaned THF ---
% 2.72/1.60 thf(ty_t_Real_Oreal,type,
% 2.72/1.60 real: $tType ).
% 2.72/1.60
% 2.72/1.60 thf(ty_t_Nat_Onat,type,
% 2.72/1.60 nat: $tType ).
% 2.72/1.60
% 2.72/1.60 thf(sy_cl_HOL_Otype,type,
% 2.72/1.60 type:
% 2.72/1.60 !>[A: $tType] : $o ).
% 2.72/1.60
% 2.72/1.60 thf(sy_cl_Groups_Oone,type,
% 2.72/1.60 one:
% 2.72/1.60 !>[A: $tType] : $o ).
% 2.72/1.60
% 2.72/1.60 thf(sy_cl_Rings_Oring,type,
% 2.72/1.60 ring:
% 2.72/1.60 !>[A: $tType] : $o ).
% 2.72/1.60
% 2.72/1.60 thf(sy_cl_Groups_Ozero,type,
% 2.72/1.60 zero:
% 2.72/1.60 !>[A: $tType] : $o ).
% 2.72/1.60
% 2.72/1.60 thf(sy_cl_Groups_Ominus,type,
% 2.72/1.60 minus:
% 2.72/1.60 !>[A: $tType] : $o ).
% 2.72/1.60
% 2.72/1.60 thf(sy_cl_Rings_Osemiring,type,
% 2.72/1.60 semiring:
% 2.72/1.60 !>[A: $tType] : $o ).
% 2.72/1.60
% 2.72/1.60 thf(sy_cl_Num_Oneg__numeral,type,
% 2.72/1.60 neg_numeral:
% 2.72/1.60 !>[A: $tType] : $o ).
% 2.72/1.60
% 2.72/1.60 thf(sy_cl_Rings_Omult__zero,type,
% 2.72/1.60 mult_zero:
% 2.72/1.60 !>[A: $tType] : $o ).
% 2.72/1.60
% 2.72/1.60 thf(sy_cl_Groups_Ogroup__add,type,
% 2.72/1.60 group_add:
% 2.72/1.60 !>[A: $tType] : $o ).
% 2.72/1.60
% 2.72/1.60 thf(sy_cl_Groups_Omonoid__add,type,
% 2.72/1.60 monoid_add:
% 2.72/1.60 !>[A: $tType] : $o ).
% 2.72/1.60
% 2.72/1.60 thf(sy_cl_Groups_Omonoid__mult,type,
% 2.72/1.60 monoid_mult:
% 2.72/1.60 !>[A: $tType] : $o ).
% 2.72/1.60
% 2.72/1.60 thf(sy_cl_Rings_Oidom__abs__sgn,type,
% 2.72/1.60 idom_abs_sgn:
% 2.72/1.60 !>[A: $tType] : $o ).
% 2.72/1.60
% 2.72/1.60 thf(sy_cl_Rings_Ozero__neq__one,type,
% 2.72/1.60 zero_neq_one:
% 2.72/1.60 !>[A: $tType] : $o ).
% 2.72/1.60
% 2.72/1.60 thf(sy_cl_Groups_Oab__group__add,type,
% 2.72/1.60 ab_group_add:
% 2.72/1.60 !>[A: $tType] : $o ).
% 2.72/1.60
% 2.72/1.60 thf(sy_cl_Rings_Ocomm__semiring,type,
% 2.72/1.60 comm_semiring:
% 2.72/1.60 !>[A: $tType] : $o ).
% 2.72/1.60
% 2.72/1.60 thf(sy_cl_Groups_Osemigroup__add,type,
% 2.72/1.60 semigroup_add:
% 2.72/1.60 !>[A: $tType] : $o ).
% 2.72/1.60
% 2.72/1.60 thf(sy_cl_Groups_Osemigroup__mult,type,
% 2.72/1.60 semigroup_mult:
% 2.72/1.60 !>[A: $tType] : $o ).
% 2.72/1.60
% 2.72/1.60 thf(sy_cl_Rings_Olinordered__idom,type,
% 2.72/1.60 linordered_idom:
% 2.72/1.60 !>[A: $tType] : $o ).
% 2.72/1.60
% 2.72/1.60 thf(sy_cl_Groups_Ocomm__monoid__add,type,
% 2.72/1.60 comm_monoid_add:
% 2.72/1.60 !>[A: $tType] : $o ).
% 2.72/1.60
% 2.72/1.60 thf(sy_cl_Rings_Oordered__semiring,type,
% 2.72/1.60 ordered_semiring:
% 2.72/1.60 !>[A: $tType] : $o ).
% 2.72/1.60
% 2.72/1.60 thf(sy_cl_Groups_Oab__semigroup__add,type,
% 2.72/1.60 ab_semigroup_add:
% 2.72/1.60 !>[A: $tType] : $o ).
% 2.72/1.60
% 2.72/1.60 thf(sy_cl_Groups_Ocomm__monoid__diff,type,
% 2.72/1.60 comm_monoid_diff:
% 2.72/1.60 !>[A: $tType] : $o ).
% 2.72/1.60
% 2.72/1.60 thf(sy_cl_Groups_Ocomm__monoid__mult,type,
% 2.72/1.60 comm_monoid_mult:
% 2.72/1.60 !>[A: $tType] : $o ).
% 2.72/1.60
% 2.72/1.60 thf(sy_cl_Groups_Oab__semigroup__mult,type,
% 2.72/1.60 ab_semigroup_mult:
% 2.72/1.60 !>[A: $tType] : $o ).
% 2.72/1.60
% 2.72/1.60 thf(sy_cl_Rings_Olinordered__semidom,type,
% 2.72/1.60 linordered_semidom:
% 2.72/1.60 !>[A: $tType] : $o ).
% 2.72/1.60
% 2.72/1.60 thf(sy_cl_Groups_Ocancel__semigroup__add,type,
% 2.72/1.60 cancel_semigroup_add:
% 2.72/1.60 !>[A: $tType] : $o ).
% 2.72/1.60
% 2.72/1.60 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 2.72/1.60 ordered_ab_group_add:
% 2.72/1.60 !>[A: $tType] : $o ).
% 2.72/1.60
% 2.72/1.60 thf(sy_cl_Rings_Ocomm__semiring__1__cancel,type,
% 2.72/1.60 comm_s1003936772cancel:
% 2.72/1.60 !>[A: $tType] : $o ).
% 2.72/1.60
% 2.72/1.60 thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
% 2.72/1.60 cancel1352612707id_add:
% 2.72/1.60 !>[A: $tType] : $o ).
% 2.72/1.60
% 2.72/1.60 thf(sy_cl_Rings_Oring__1__no__zero__divisors,type,
% 2.72/1.60 ring_11004092258visors:
% 2.72/1.60 !>[A: $tType] : $o ).
% 2.72/1.60
% 2.72/1.60 thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
% 2.72/1.60 cancel146912293up_add:
% 2.72/1.60 !>[A: $tType] : $o ).
% 2.72/1.60
% 2.72/1.60 thf(sy_cl_Groups_Olinordered__ab__group__add,type,
% 2.72/1.60 linord219039673up_add:
% 2.72/1.60 !>[A: $tType] : $o ).
% 2.72/1.60
% 2.72/1.60 thf(sy_cl_Groups_Oordered__comm__monoid__add,type,
% 2.72/1.60 ordere216010020id_add:
% 2.72/1.60 !>[A: $tType] : $o ).
% 2.72/1.60
% 2.72/1.60 thf(sy_cl_Groups_Oordered__ab__group__add__abs,type,
% 2.72/1.60 ordere142940540dd_abs:
% 2.72/1.60 !>[A: $tType] : $o ).
% 2.72/1.60
% 2.72/1.60 thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
% 2.72/1.60 ordere779506340up_add:
% 2.72/1.60 !>[A: $tType] : $o ).
% 2.72/1.60
% 2.72/1.60 thf(sy_cl_Rings_Osemiring__no__zero__divisors,type,
% 2.72/1.60 semiri1193490041visors:
% 2.72/1.60 !>[A: $tType] : $o ).
% 2.72/1.60
% 2.72/1.60 thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
% 2.72/1.60 linord1659791738miring:
% 2.72/1.60 !>[A: $tType] : $o ).
% 2.72/1.60
% 2.72/1.60 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 2.72/1.60 canoni770627133id_add:
% 2.72/1.60 !>[A: $tType] : $o ).
% 2.72/1.60
% 2.72/1.60 thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
% 2.72/1.60 ordere236663937imp_le:
% 2.72/1.60 !>[A: $tType] : $o ).
% 2.72/1.60
% 2.72/1.60 thf(sy_cl_Groups_Oordered__cancel__comm__monoid__diff,type,
% 2.72/1.60 ordere623563068d_diff:
% 2.72/1.60 !>[A: $tType] : $o ).
% 2.72/1.60
% 2.72/1.60 thf(sy_cl_Rings_Osemiring__no__zero__divisors__cancel,type,
% 2.72/1.60 semiri1923998003cancel:
% 2.72/1.60 !>[A: $tType] : $o ).
% 2.72/1.60
% 2.72/1.60 thf(sy_cl_Groups_Oordered__ab__semigroup__monoid__add__imp__le,type,
% 2.72/1.60 ordere516151231imp_le:
% 2.72/1.60 !>[A: $tType] : $o ).
% 2.72/1.60
% 2.72/1.60 thf(sy_c_GenClock__Mirabelle__pukziqipvs_OPC,type,
% 2.72/1.60 genClo1161277105lle_PC: nat > real > real ).
% 2.72/1.60
% 2.72/1.60 thf(sy_c_GenClock__Mirabelle__pukziqipvs_O_092_060beta_062,type,
% 2.72/1.60 genClo1278781456e_beta: real ).
% 2.72/1.60
% 2.72/1.60 thf(sy_c_GenClock__Mirabelle__pukziqipvs_O_092_060mu_062,type,
% 2.72/1.60 genClo872980712lle_mu: real ).
% 2.72/1.60
% 2.72/1.60 thf(sy_c_GenClock__Mirabelle__pukziqipvs_O_092_060rho_062,type,
% 2.72/1.60 genClo1144207539le_rho: real ).
% 2.72/1.60
% 2.72/1.60 thf(sy_c_GenClock__Mirabelle__pukziqipvs_Ocorrect,type,
% 2.72/1.60 genClo1015804716orrect: nat > real > $o ).
% 2.72/1.60
% 2.72/1.60 thf(sy_c_GenClock__Mirabelle__pukziqipvs_OokRead2,type,
% 2.72/1.60 genClo293725282kRead2: ( nat > real ) > ( nat > real ) > real > ( nat > $o ) > $o ).
% 2.72/1.60
% 2.72/1.60 thf(sy_c_GenClock__Mirabelle__pukziqipvs_Ormax,type,
% 2.72/1.60 genClo1650508560e_rmax: real ).
% 2.72/1.60
% 2.72/1.60 thf(sy_c_GenClock__Mirabelle__pukziqipvs_Ormin,type,
% 2.72/1.60 genClo1651033342e_rmin: real ).
% 2.72/1.60
% 2.72/1.60 thf(sy_c_GenClock__Mirabelle__pukziqipvs_Ote,type,
% 2.72/1.60 genClo1163638703lle_te: nat > nat > real ).
% 2.72/1.60
% 2.72/1.60 thf(sy_c_Groups_Oabs__class_Oabs,type,
% 2.72/1.60 abs_abs:
% 2.72/1.60 !>[A: $tType] : ( A > A ) ).
% 2.72/1.60
% 2.72/1.60 thf(sy_c_Groups_Ominus__class_Ominus,type,
% 2.72/1.60 minus_minus:
% 2.72/1.60 !>[A: $tType] : ( A > A > A ) ).
% 2.72/1.60
% 2.72/1.60 thf(sy_c_Groups_Oone__class_Oone,type,
% 2.72/1.60 one_one:
% 2.72/1.60 !>[A: $tType] : A ).
% 2.72/1.60
% 2.72/1.60 thf(sy_c_Groups_Oplus__class_Oplus,type,
% 2.72/1.60 plus_plus:
% 2.72/1.60 !>[A: $tType] : ( A > A > A ) ).
% 2.72/1.60
% 2.72/1.60 thf(sy_c_Groups_Otimes__class_Otimes,type,
% 2.72/1.60 times_times:
% 2.72/1.60 !>[A: $tType] : ( A > A > A ) ).
% 2.72/1.60
% 2.72/1.60 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 2.72/1.60 zero_zero:
% 2.72/1.60 !>[A: $tType] : A ).
% 2.72/1.60
% 2.72/1.60 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 2.72/1.60 ord_less_eq:
% 2.72/1.60 !>[A: $tType] : ( A > A > $o ) ).
% 2.72/1.60
% 2.72/1.60 thf(sy_v_i,type,
% 2.72/1.60 i: nat ).
% 2.72/1.60
% 2.72/1.60 thf(sy_v_l,type,
% 2.72/1.60 l: nat ).
% 2.72/1.60
% 2.72/1.60 thf(sy_v_p,type,
% 2.72/1.60 p: nat ).
% 2.72/1.60
% 2.72/1.60 thf(sy_v_q,type,
% 2.72/1.60 q: nat ).
% 2.72/1.60
% 2.72/1.60 thf(fact_0_rts2b,axiom,
% 2.72/1.60 ! [P: nat,Q: nat,I: nat] :
% 2.72/1.60 ( ( ( genClo1015804716orrect @ P @ ( genClo1163638703lle_te @ P @ I ) )
% 2.72/1.60 & ( genClo1015804716orrect @ Q @ ( genClo1163638703lle_te @ Q @ I ) ) )
% 2.72/1.60 => ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( genClo1163638703lle_te @ P @ I ) @ ( genClo1163638703lle_te @ Q @ I ) ) ) @ genClo1278781456e_beta ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_1_corr__p,axiom,
% 2.72/1.60 genClo1015804716orrect @ p @ ( genClo1163638703lle_te @ p @ ( plus_plus @ nat @ i @ ( one_one @ nat ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_2_corr__q__tq,axiom,
% 2.72/1.60 genClo1015804716orrect @ q @ ( genClo1163638703lle_te @ q @ ( plus_plus @ nat @ i @ ( one_one @ nat ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_3_ie,axiom,
% 2.72/1.60 ord_less_eq @ real @ ( genClo1163638703lle_te @ q @ ( plus_plus @ nat @ i @ ( one_one @ nat ) ) ) @ ( genClo1163638703lle_te @ p @ ( plus_plus @ nat @ i @ ( one_one @ nat ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_4_corr__q,axiom,
% 2.72/1.60 genClo1015804716orrect @ q @ ( genClo1163638703lle_te @ p @ ( plus_plus @ nat @ i @ ( one_one @ nat ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_5_corr__l,axiom,
% 2.72/1.60 genClo1015804716orrect @ l @ ( genClo1163638703lle_te @ p @ ( plus_plus @ nat @ i @ ( one_one @ nat ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_6_posD,axiom,
% 2.72/1.60 ord_less_eq @ real @ ( zero_zero @ real ) @ ( minus_minus @ real @ ( genClo1163638703lle_te @ p @ ( plus_plus @ nat @ i @ ( one_one @ nat ) ) ) @ ( genClo1163638703lle_te @ q @ ( plus_plus @ nat @ i @ ( one_one @ nat ) ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_7_corr__l__tq,axiom,
% 2.72/1.60 genClo1015804716orrect @ l @ ( genClo1163638703lle_te @ q @ ( plus_plus @ nat @ i @ ( one_one @ nat ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_8_le__add__diff__inverse,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( linordered_semidom @ A )
% 2.72/1.60 => ! [B: A,A2: A] :
% 2.72/1.60 ( ( ord_less_eq @ A @ B @ A2 )
% 2.72/1.60 => ( ( plus_plus @ A @ B @ ( minus_minus @ A @ A2 @ B ) )
% 2.72/1.60 = A2 ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_9_le__add__diff__inverse2,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( linordered_semidom @ A )
% 2.72/1.60 => ! [B: A,A2: A] :
% 2.72/1.60 ( ( ord_less_eq @ A @ B @ A2 )
% 2.72/1.60 => ( ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B ) @ B )
% 2.72/1.60 = A2 ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_10_rte,axiom,
% 2.72/1.60 ! [P2: nat,I2: nat] :
% 2.72/1.60 ( ( genClo1015804716orrect @ P2 @ ( genClo1163638703lle_te @ P2 @ ( plus_plus @ nat @ I2 @ ( one_one @ nat ) ) ) )
% 2.72/1.60 => ( ord_less_eq @ real @ ( genClo1163638703lle_te @ P2 @ I2 ) @ ( genClo1163638703lle_te @ P2 @ ( plus_plus @ nat @ I2 @ ( one_one @ nat ) ) ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_11_abs__1,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( idom_abs_sgn @ A )
% 2.72/1.60 => ( ( abs_abs @ A @ ( one_one @ A ) )
% 2.72/1.60 = ( one_one @ A ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_12_abs__add__abs,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( ordere142940540dd_abs @ A )
% 2.72/1.60 => ! [A2: A,B: A] :
% 2.72/1.60 ( ( abs_abs @ A @ ( plus_plus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B ) ) )
% 2.72/1.60 = ( plus_plus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B ) ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_13_add__diff__cancel,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( group_add @ A )
% 2.72/1.60 => ! [A2: A,B: A] :
% 2.72/1.60 ( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B ) @ B )
% 2.72/1.60 = A2 ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_14_diff__add__cancel,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( group_add @ A )
% 2.72/1.60 => ! [A2: A,B: A] :
% 2.72/1.60 ( ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B ) @ B )
% 2.72/1.60 = A2 ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_15_add__diff__cancel__left,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( cancel146912293up_add @ A )
% 2.72/1.60 => ! [C: A,A2: A,B: A] :
% 2.72/1.60 ( ( minus_minus @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) )
% 2.72/1.60 = ( minus_minus @ A @ A2 @ B ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_16_add__diff__cancel__left_H,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( cancel146912293up_add @ A )
% 2.72/1.60 => ! [A2: A,B: A] :
% 2.72/1.60 ( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B ) @ A2 )
% 2.72/1.60 = B ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_17_add__diff__cancel__right,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( cancel146912293up_add @ A )
% 2.72/1.60 => ! [A2: A,C: A,B: A] :
% 2.72/1.60 ( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) )
% 2.72/1.60 = ( minus_minus @ A @ A2 @ B ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_18_add__right__cancel,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( cancel_semigroup_add @ A )
% 2.72/1.60 => ! [B: A,A2: A,C: A] :
% 2.72/1.60 ( ( ( plus_plus @ A @ B @ A2 )
% 2.72/1.60 = ( plus_plus @ A @ C @ A2 ) )
% 2.72/1.60 = ( B = C ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_19_add__left__cancel,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( cancel_semigroup_add @ A )
% 2.72/1.60 => ! [A2: A,B: A,C: A] :
% 2.72/1.60 ( ( ( plus_plus @ A @ A2 @ B )
% 2.72/1.60 = ( plus_plus @ A @ A2 @ C ) )
% 2.72/1.60 = ( B = C ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_20_abs__idempotent,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( ordere142940540dd_abs @ A )
% 2.72/1.60 => ! [A2: A] :
% 2.72/1.60 ( ( abs_abs @ A @ ( abs_abs @ A @ A2 ) )
% 2.72/1.60 = ( abs_abs @ A @ A2 ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_21_abs__abs,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( idom_abs_sgn @ A )
% 2.72/1.60 => ! [A2: A] :
% 2.72/1.60 ( ( abs_abs @ A @ ( abs_abs @ A @ A2 ) )
% 2.72/1.60 = ( abs_abs @ A @ A2 ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_22_le__zero__eq,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( canoni770627133id_add @ A )
% 2.72/1.60 => ! [N: A] :
% 2.72/1.60 ( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
% 2.72/1.60 = ( N
% 2.72/1.60 = ( zero_zero @ A ) ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_23_add__le__cancel__right,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( ordere236663937imp_le @ A )
% 2.72/1.60 => ! [A2: A,C: A,B: A] :
% 2.72/1.60 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) )
% 2.72/1.60 = ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_24_add__le__cancel__left,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( ordere236663937imp_le @ A )
% 2.72/1.60 => ! [C: A,A2: A,B: A] :
% 2.72/1.60 ( ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) )
% 2.72/1.60 = ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_25_zero__eq__add__iff__both__eq__0,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( canoni770627133id_add @ A )
% 2.72/1.60 => ! [X: A,Y: A] :
% 2.72/1.60 ( ( ( zero_zero @ A )
% 2.72/1.60 = ( plus_plus @ A @ X @ Y ) )
% 2.72/1.60 = ( ( X
% 2.72/1.60 = ( zero_zero @ A ) )
% 2.72/1.60 & ( Y
% 2.72/1.60 = ( zero_zero @ A ) ) ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_26_add__eq__0__iff__both__eq__0,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( canoni770627133id_add @ A )
% 2.72/1.60 => ! [X: A,Y: A] :
% 2.72/1.60 ( ( ( plus_plus @ A @ X @ Y )
% 2.72/1.60 = ( zero_zero @ A ) )
% 2.72/1.60 = ( ( X
% 2.72/1.60 = ( zero_zero @ A ) )
% 2.72/1.60 & ( Y
% 2.72/1.60 = ( zero_zero @ A ) ) ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_27_add__cancel__right__right,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( cancel1352612707id_add @ A )
% 2.72/1.60 => ! [A2: A,B: A] :
% 2.72/1.60 ( ( A2
% 2.72/1.60 = ( plus_plus @ A @ A2 @ B ) )
% 2.72/1.60 = ( B
% 2.72/1.60 = ( zero_zero @ A ) ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_28_add__cancel__right__left,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( cancel1352612707id_add @ A )
% 2.72/1.60 => ! [A2: A,B: A] :
% 2.72/1.60 ( ( A2
% 2.72/1.60 = ( plus_plus @ A @ B @ A2 ) )
% 2.72/1.60 = ( B
% 2.72/1.60 = ( zero_zero @ A ) ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_29_add__cancel__left__right,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( cancel1352612707id_add @ A )
% 2.72/1.60 => ! [A2: A,B: A] :
% 2.72/1.60 ( ( ( plus_plus @ A @ A2 @ B )
% 2.72/1.60 = A2 )
% 2.72/1.60 = ( B
% 2.72/1.60 = ( zero_zero @ A ) ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_30_add__cancel__left__left,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( cancel1352612707id_add @ A )
% 2.72/1.60 => ! [B: A,A2: A] :
% 2.72/1.60 ( ( ( plus_plus @ A @ B @ A2 )
% 2.72/1.60 = A2 )
% 2.72/1.60 = ( B
% 2.72/1.60 = ( zero_zero @ A ) ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_31_double__zero__sym,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( linord219039673up_add @ A )
% 2.72/1.60 => ! [A2: A] :
% 2.72/1.60 ( ( ( zero_zero @ A )
% 2.72/1.60 = ( plus_plus @ A @ A2 @ A2 ) )
% 2.72/1.60 = ( A2
% 2.72/1.60 = ( zero_zero @ A ) ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_32_double__zero,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( linord219039673up_add @ A )
% 2.72/1.60 => ! [A2: A] :
% 2.72/1.60 ( ( ( plus_plus @ A @ A2 @ A2 )
% 2.72/1.60 = ( zero_zero @ A ) )
% 2.72/1.60 = ( A2
% 2.72/1.60 = ( zero_zero @ A ) ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_33_add_Oright__neutral,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( monoid_add @ A )
% 2.72/1.60 => ! [A2: A] :
% 2.72/1.60 ( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
% 2.72/1.60 = A2 ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_34_add_Oleft__neutral,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( monoid_add @ A )
% 2.72/1.60 => ! [A2: A] :
% 2.72/1.60 ( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
% 2.72/1.60 = A2 ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_35_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( cancel1352612707id_add @ A )
% 2.72/1.60 => ! [A2: A] :
% 2.72/1.60 ( ( minus_minus @ A @ A2 @ A2 )
% 2.72/1.60 = ( zero_zero @ A ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_36_diff__zero,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( cancel1352612707id_add @ A )
% 2.72/1.60 => ! [A2: A] :
% 2.72/1.60 ( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
% 2.72/1.60 = A2 ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_37_zero__diff,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( comm_monoid_diff @ A )
% 2.72/1.60 => ! [A2: A] :
% 2.72/1.60 ( ( minus_minus @ A @ ( zero_zero @ A ) @ A2 )
% 2.72/1.60 = ( zero_zero @ A ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_38_diff__0__right,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( group_add @ A )
% 2.72/1.60 => ! [A2: A] :
% 2.72/1.60 ( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
% 2.72/1.60 = A2 ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_39_diff__self,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( group_add @ A )
% 2.72/1.60 => ! [A2: A] :
% 2.72/1.60 ( ( minus_minus @ A @ A2 @ A2 )
% 2.72/1.60 = ( zero_zero @ A ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_40_add__diff__cancel__right_H,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( cancel146912293up_add @ A )
% 2.72/1.60 => ! [A2: A,B: A] :
% 2.72/1.60 ( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B ) @ B )
% 2.72/1.60 = A2 ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_41_abs__zero,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( ordere142940540dd_abs @ A )
% 2.72/1.60 => ( ( abs_abs @ A @ ( zero_zero @ A ) )
% 2.72/1.60 = ( zero_zero @ A ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_42_abs__eq__0,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( ordere142940540dd_abs @ A )
% 2.72/1.60 => ! [A2: A] :
% 2.72/1.60 ( ( ( abs_abs @ A @ A2 )
% 2.72/1.60 = ( zero_zero @ A ) )
% 2.72/1.60 = ( A2
% 2.72/1.60 = ( zero_zero @ A ) ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_43_ext,axiom,
% 2.72/1.60 ! [B2: $tType,A: $tType,F: A > B2,G: A > B2] :
% 2.72/1.60 ( ! [X2: A] :
% 2.72/1.60 ( ( F @ X2 )
% 2.72/1.60 = ( G @ X2 ) )
% 2.72/1.60 => ( F = G ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_44_abs__0__eq,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( ordere142940540dd_abs @ A )
% 2.72/1.60 => ! [A2: A] :
% 2.72/1.60 ( ( ( zero_zero @ A )
% 2.72/1.60 = ( abs_abs @ A @ A2 ) )
% 2.72/1.60 = ( A2
% 2.72/1.60 = ( zero_zero @ A ) ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_45_abs__0,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( idom_abs_sgn @ A )
% 2.72/1.60 => ( ( abs_abs @ A @ ( zero_zero @ A ) )
% 2.72/1.60 = ( zero_zero @ A ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_46_zero__le__double__add__iff__zero__le__single__add,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( linord219039673up_add @ A )
% 2.72/1.60 => ! [A2: A] :
% 2.72/1.60 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ A2 ) )
% 2.72/1.60 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_47_double__add__le__zero__iff__single__add__le__zero,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( linord219039673up_add @ A )
% 2.72/1.60 => ! [A2: A] :
% 2.72/1.60 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ A2 ) @ ( zero_zero @ A ) )
% 2.72/1.60 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_48_le__add__same__cancel2,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( ordere516151231imp_le @ A )
% 2.72/1.60 => ! [A2: A,B: A] :
% 2.72/1.60 ( ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ B @ A2 ) )
% 2.72/1.60 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ B ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_49_le__add__same__cancel1,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( ordere516151231imp_le @ A )
% 2.72/1.60 => ! [A2: A,B: A] :
% 2.72/1.60 ( ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ A2 @ B ) )
% 2.72/1.60 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ B ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_50_add__le__same__cancel2,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( ordere516151231imp_le @ A )
% 2.72/1.60 => ! [A2: A,B: A] :
% 2.72/1.60 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B ) @ B )
% 2.72/1.60 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_51_add__le__same__cancel1,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( ordere516151231imp_le @ A )
% 2.72/1.60 => ! [B: A,A2: A] :
% 2.72/1.60 ( ( ord_less_eq @ A @ ( plus_plus @ A @ B @ A2 ) @ B )
% 2.72/1.60 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_52_diff__ge__0__iff__ge,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( ordered_ab_group_add @ A )
% 2.72/1.60 => ! [A2: A,B: A] :
% 2.72/1.60 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A2 @ B ) )
% 2.72/1.60 = ( ord_less_eq @ A @ B @ A2 ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_53_diff__add__zero,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( comm_monoid_diff @ A )
% 2.72/1.60 => ! [A2: A,B: A] :
% 2.72/1.60 ( ( minus_minus @ A @ A2 @ ( plus_plus @ A @ A2 @ B ) )
% 2.72/1.60 = ( zero_zero @ A ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_54_abs__le__zero__iff,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( ordere142940540dd_abs @ A )
% 2.72/1.60 => ! [A2: A] :
% 2.72/1.60 ( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ ( zero_zero @ A ) )
% 2.72/1.60 = ( A2
% 2.72/1.60 = ( zero_zero @ A ) ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_55_abs__le__self__iff,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( ordere142940540dd_abs @ A )
% 2.72/1.60 => ! [A2: A] :
% 2.72/1.60 ( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ A2 )
% 2.72/1.60 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_56_abs__of__nonneg,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( ordere142940540dd_abs @ A )
% 2.72/1.60 => ! [A2: A] :
% 2.72/1.60 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 2.72/1.60 => ( ( abs_abs @ A @ A2 )
% 2.72/1.60 = A2 ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_57_zero__reorient,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( zero @ A )
% 2.72/1.60 => ! [X: A] :
% 2.72/1.60 ( ( ( zero_zero @ A )
% 2.72/1.60 = X )
% 2.72/1.60 = ( X
% 2.72/1.60 = ( zero_zero @ A ) ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_58_zero__le,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( canoni770627133id_add @ A )
% 2.72/1.60 => ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_59_add_Ogroup__left__neutral,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( group_add @ A )
% 2.72/1.60 => ! [A2: A] :
% 2.72/1.60 ( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
% 2.72/1.60 = A2 ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_60_add_Ocomm__neutral,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( comm_monoid_add @ A )
% 2.72/1.60 => ! [A2: A] :
% 2.72/1.60 ( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
% 2.72/1.60 = A2 ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_61_comm__monoid__add__class_Oadd__0,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( comm_monoid_add @ A )
% 2.72/1.60 => ! [A2: A] :
% 2.72/1.60 ( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
% 2.72/1.60 = A2 ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_62_zero__neq__one,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( zero_neq_one @ A )
% 2.72/1.60 => ( ( zero_zero @ A )
% 2.72/1.60 != ( one_one @ A ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_63_eq__iff__diff__eq__0,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( group_add @ A )
% 2.72/1.60 => ( ( ^ [Y2: A,Z: A] : ( Y2 = Z ) )
% 2.72/1.60 = ( ^ [A3: A,B3: A] :
% 2.72/1.60 ( ( minus_minus @ A @ A3 @ B3 )
% 2.72/1.60 = ( zero_zero @ A ) ) ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_64_abs__eq__0__iff,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( idom_abs_sgn @ A )
% 2.72/1.60 => ! [A2: A] :
% 2.72/1.60 ( ( ( abs_abs @ A @ A2 )
% 2.72/1.60 = ( zero_zero @ A ) )
% 2.72/1.60 = ( A2
% 2.72/1.60 = ( zero_zero @ A ) ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_65_add__nonpos__eq__0__iff,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( ordere216010020id_add @ A )
% 2.72/1.60 => ! [X: A,Y: A] :
% 2.72/1.60 ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
% 2.72/1.60 => ( ( ord_less_eq @ A @ Y @ ( zero_zero @ A ) )
% 2.72/1.60 => ( ( ( plus_plus @ A @ X @ Y )
% 2.72/1.60 = ( zero_zero @ A ) )
% 2.72/1.60 = ( ( X
% 2.72/1.60 = ( zero_zero @ A ) )
% 2.72/1.60 & ( Y
% 2.72/1.60 = ( zero_zero @ A ) ) ) ) ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_66_add__nonneg__eq__0__iff,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( ordere216010020id_add @ A )
% 2.72/1.60 => ! [X: A,Y: A] :
% 2.72/1.60 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 2.72/1.60 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
% 2.72/1.60 => ( ( ( plus_plus @ A @ X @ Y )
% 2.72/1.60 = ( zero_zero @ A ) )
% 2.72/1.60 = ( ( X
% 2.72/1.60 = ( zero_zero @ A ) )
% 2.72/1.60 & ( Y
% 2.72/1.60 = ( zero_zero @ A ) ) ) ) ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_67_add__nonpos__nonpos,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( ordere216010020id_add @ A )
% 2.72/1.60 => ! [A2: A,B: A] :
% 2.72/1.60 ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
% 2.72/1.60 => ( ( ord_less_eq @ A @ B @ ( zero_zero @ A ) )
% 2.72/1.60 => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B ) @ ( zero_zero @ A ) ) ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_68_add__nonneg__nonneg,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( ordere216010020id_add @ A )
% 2.72/1.60 => ! [A2: A,B: A] :
% 2.72/1.60 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 2.72/1.60 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B )
% 2.72/1.60 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B ) ) ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_69_add__increasing2,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( ordere216010020id_add @ A )
% 2.72/1.60 => ! [C: A,B: A,A2: A] :
% 2.72/1.60 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C )
% 2.72/1.60 => ( ( ord_less_eq @ A @ B @ A2 )
% 2.72/1.60 => ( ord_less_eq @ A @ B @ ( plus_plus @ A @ A2 @ C ) ) ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_70_add__decreasing2,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( ordere216010020id_add @ A )
% 2.72/1.60 => ! [C: A,A2: A,B: A] :
% 2.72/1.60 ( ( ord_less_eq @ A @ C @ ( zero_zero @ A ) )
% 2.72/1.60 => ( ( ord_less_eq @ A @ A2 @ B )
% 2.72/1.60 => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ B ) ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_71_add__increasing,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( ordere216010020id_add @ A )
% 2.72/1.60 => ! [A2: A,B: A,C: A] :
% 2.72/1.60 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 2.72/1.60 => ( ( ord_less_eq @ A @ B @ C )
% 2.72/1.60 => ( ord_less_eq @ A @ B @ ( plus_plus @ A @ A2 @ C ) ) ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_72_add__decreasing,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( ordere216010020id_add @ A )
% 2.72/1.60 => ! [A2: A,C: A,B: A] :
% 2.72/1.60 ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
% 2.72/1.60 => ( ( ord_less_eq @ A @ C @ B )
% 2.72/1.60 => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ B ) ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_73_not__one__le__zero,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( linord1659791738miring @ A )
% 2.72/1.60 => ~ ( ord_less_eq @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_74_zero__le__one,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( linord1659791738miring @ A )
% 2.72/1.60 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_75_le__iff__diff__le__0,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( ordered_ab_group_add @ A )
% 2.72/1.60 => ( ( ord_less_eq @ A )
% 2.72/1.60 = ( ^ [A3: A,B3: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ B3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_76_abs__ge__zero,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( ordere142940540dd_abs @ A )
% 2.72/1.60 => ! [A2: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( abs_abs @ A @ A2 ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_77_rts2a,axiom,
% 2.72/1.60 ! [P: nat,Q: nat,T: real,I: nat] :
% 2.72/1.60 ( ( ( genClo1015804716orrect @ P @ T )
% 2.72/1.60 & ( genClo1015804716orrect @ Q @ T )
% 2.72/1.60 & ( ord_less_eq @ real @ ( plus_plus @ real @ ( genClo1163638703lle_te @ Q @ I ) @ genClo1278781456e_beta ) @ T ) )
% 2.72/1.60 => ( ord_less_eq @ real @ ( genClo1163638703lle_te @ P @ I ) @ T ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_78_add__right__imp__eq,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( cancel_semigroup_add @ A )
% 2.72/1.60 => ! [B: A,A2: A,C: A] :
% 2.72/1.60 ( ( ( plus_plus @ A @ B @ A2 )
% 2.72/1.60 = ( plus_plus @ A @ C @ A2 ) )
% 2.72/1.60 => ( B = C ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_79_add__left__imp__eq,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( cancel_semigroup_add @ A )
% 2.72/1.60 => ! [A2: A,B: A,C: A] :
% 2.72/1.60 ( ( ( plus_plus @ A @ A2 @ B )
% 2.72/1.60 = ( plus_plus @ A @ A2 @ C ) )
% 2.72/1.60 => ( B = C ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_80_add_Oleft__commute,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( ab_semigroup_add @ A )
% 2.72/1.60 => ! [B: A,A2: A,C: A] :
% 2.72/1.60 ( ( plus_plus @ A @ B @ ( plus_plus @ A @ A2 @ C ) )
% 2.72/1.60 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_81_add_Ocommute,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( ab_semigroup_add @ A )
% 2.72/1.60 => ( ( plus_plus @ A )
% 2.72/1.60 = ( ^ [A3: A,B3: A] : ( plus_plus @ A @ B3 @ A3 ) ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_82_add_Oright__cancel,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( group_add @ A )
% 2.72/1.60 => ! [B: A,A2: A,C: A] :
% 2.72/1.60 ( ( ( plus_plus @ A @ B @ A2 )
% 2.72/1.60 = ( plus_plus @ A @ C @ A2 ) )
% 2.72/1.60 = ( B = C ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_83_add_Oleft__cancel,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( group_add @ A )
% 2.72/1.60 => ! [A2: A,B: A,C: A] :
% 2.72/1.60 ( ( ( plus_plus @ A @ A2 @ B )
% 2.72/1.60 = ( plus_plus @ A @ A2 @ C ) )
% 2.72/1.60 = ( B = C ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_84_add_Oassoc,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( semigroup_add @ A )
% 2.72/1.60 => ! [A2: A,B: A,C: A] :
% 2.72/1.60 ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B ) @ C )
% 2.72/1.60 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_85_group__cancel_Oadd2,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( comm_monoid_add @ A )
% 2.72/1.60 => ! [B4: A,K: A,B: A,A2: A] :
% 2.72/1.60 ( ( B4
% 2.72/1.60 = ( plus_plus @ A @ K @ B ) )
% 2.72/1.60 => ( ( plus_plus @ A @ A2 @ B4 )
% 2.72/1.60 = ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B ) ) ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_86_group__cancel_Oadd1,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( comm_monoid_add @ A )
% 2.72/1.60 => ! [A4: A,K: A,A2: A,B: A] :
% 2.72/1.60 ( ( A4
% 2.72/1.60 = ( plus_plus @ A @ K @ A2 ) )
% 2.72/1.60 => ( ( plus_plus @ A @ A4 @ B )
% 2.72/1.60 = ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B ) ) ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_87_add__mono__thms__linordered__semiring_I4_J,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( ordere779506340up_add @ A )
% 2.72/1.60 => ! [I2: A,J: A,K: A,L: A] :
% 2.72/1.60 ( ( ( I2 = J )
% 2.72/1.60 & ( K = L ) )
% 2.72/1.60 => ( ( plus_plus @ A @ I2 @ K )
% 2.72/1.60 = ( plus_plus @ A @ J @ L ) ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_88_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( ab_semigroup_add @ A )
% 2.72/1.60 => ! [A2: A,B: A,C: A] :
% 2.72/1.60 ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B ) @ C )
% 2.72/1.60 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_89_one__reorient,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( one @ A )
% 2.72/1.60 => ! [X: A] :
% 2.72/1.60 ( ( ( one_one @ A )
% 2.72/1.60 = X )
% 2.72/1.60 = ( X
% 2.72/1.60 = ( one_one @ A ) ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_90_diff__right__commute,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( cancel146912293up_add @ A )
% 2.72/1.60 => ! [A2: A,C: A,B: A] :
% 2.72/1.60 ( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C ) @ B )
% 2.72/1.60 = ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B ) @ C ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_91_diff__eq__diff__eq,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( group_add @ A )
% 2.72/1.60 => ! [A2: A,B: A,C: A,D: A] :
% 2.72/1.60 ( ( ( minus_minus @ A @ A2 @ B )
% 2.72/1.60 = ( minus_minus @ A @ C @ D ) )
% 2.72/1.60 => ( ( A2 = B )
% 2.72/1.60 = ( C = D ) ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_92_beta__bound1,axiom,
% 2.72/1.60 ! [P2: nat,I2: nat,Q2: nat] :
% 2.72/1.60 ( ( genClo1015804716orrect @ P2 @ ( genClo1163638703lle_te @ P2 @ ( plus_plus @ nat @ I2 @ ( one_one @ nat ) ) ) )
% 2.72/1.60 => ( ( genClo1015804716orrect @ Q2 @ ( genClo1163638703lle_te @ P2 @ ( plus_plus @ nat @ I2 @ ( one_one @ nat ) ) ) )
% 2.72/1.60 => ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( minus_minus @ real @ ( genClo1163638703lle_te @ P2 @ ( plus_plus @ nat @ I2 @ ( one_one @ nat ) ) ) @ ( genClo1163638703lle_te @ Q2 @ I2 ) ) ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_93_correct__closed,axiom,
% 2.72/1.60 ! [P: nat,S: real,T: real] :
% 2.72/1.60 ( ( ( ord_less_eq @ real @ S @ T )
% 2.72/1.60 & ( genClo1015804716orrect @ P @ T ) )
% 2.72/1.60 => ( genClo1015804716orrect @ P @ S ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_94_add__le__imp__le__right,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( ordere236663937imp_le @ A )
% 2.72/1.60 => ! [A2: A,C: A,B: A] :
% 2.72/1.60 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) )
% 2.72/1.60 => ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_95_add__le__imp__le__left,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( ordere236663937imp_le @ A )
% 2.72/1.60 => ! [C: A,A2: A,B: A] :
% 2.72/1.60 ( ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) )
% 2.72/1.60 => ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_96_le__iff__add,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( canoni770627133id_add @ A )
% 2.72/1.60 => ( ( ord_less_eq @ A )
% 2.72/1.60 = ( ^ [A3: A,B3: A] :
% 2.72/1.60 ? [C2: A] :
% 2.72/1.60 ( B3
% 2.72/1.60 = ( plus_plus @ A @ A3 @ C2 ) ) ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_97_add__right__mono,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( ordere779506340up_add @ A )
% 2.72/1.60 => ! [A2: A,B: A,C: A] :
% 2.72/1.60 ( ( ord_less_eq @ A @ A2 @ B )
% 2.72/1.60 => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_98_less__eqE,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( canoni770627133id_add @ A )
% 2.72/1.60 => ! [A2: A,B: A] :
% 2.72/1.60 ( ( ord_less_eq @ A @ A2 @ B )
% 2.72/1.60 => ~ ! [C3: A] :
% 2.72/1.60 ( B
% 2.72/1.60 != ( plus_plus @ A @ A2 @ C3 ) ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_99_add__left__mono,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( ordere779506340up_add @ A )
% 2.72/1.60 => ! [A2: A,B: A,C: A] :
% 2.72/1.60 ( ( ord_less_eq @ A @ A2 @ B )
% 2.72/1.60 => ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_100_add__mono,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( ordere779506340up_add @ A )
% 2.72/1.60 => ! [A2: A,B: A,C: A,D: A] :
% 2.72/1.60 ( ( ord_less_eq @ A @ A2 @ B )
% 2.72/1.60 => ( ( ord_less_eq @ A @ C @ D )
% 2.72/1.60 => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ D ) ) ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_101_add__mono__thms__linordered__semiring_I1_J,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( ordere779506340up_add @ A )
% 2.72/1.60 => ! [I2: A,J: A,K: A,L: A] :
% 2.72/1.60 ( ( ( ord_less_eq @ A @ I2 @ J )
% 2.72/1.60 & ( ord_less_eq @ A @ K @ L ) )
% 2.72/1.60 => ( ord_less_eq @ A @ ( plus_plus @ A @ I2 @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_102_add__mono__thms__linordered__semiring_I2_J,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( ordere779506340up_add @ A )
% 2.72/1.60 => ! [I2: A,J: A,K: A,L: A] :
% 2.72/1.60 ( ( ( I2 = J )
% 2.72/1.60 & ( ord_less_eq @ A @ K @ L ) )
% 2.72/1.60 => ( ord_less_eq @ A @ ( plus_plus @ A @ I2 @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_103_add__mono__thms__linordered__semiring_I3_J,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( ordere779506340up_add @ A )
% 2.72/1.60 => ! [I2: A,J: A,K: A,L: A] :
% 2.72/1.60 ( ( ( ord_less_eq @ A @ I2 @ J )
% 2.72/1.60 & ( K = L ) )
% 2.72/1.60 => ( ord_less_eq @ A @ ( plus_plus @ A @ I2 @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_104_diff__eq__diff__less__eq,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( ordered_ab_group_add @ A )
% 2.72/1.60 => ! [A2: A,B: A,C: A,D: A] :
% 2.72/1.60 ( ( ( minus_minus @ A @ A2 @ B )
% 2.72/1.60 = ( minus_minus @ A @ C @ D ) )
% 2.72/1.60 => ( ( ord_less_eq @ A @ A2 @ B )
% 2.72/1.60 = ( ord_less_eq @ A @ C @ D ) ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_105_diff__right__mono,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( ordered_ab_group_add @ A )
% 2.72/1.60 => ! [A2: A,B: A,C: A] :
% 2.72/1.60 ( ( ord_less_eq @ A @ A2 @ B )
% 2.72/1.60 => ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C ) @ ( minus_minus @ A @ B @ C ) ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_106_diff__left__mono,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( ordered_ab_group_add @ A )
% 2.72/1.60 => ! [B: A,A2: A,C: A] :
% 2.72/1.60 ( ( ord_less_eq @ A @ B @ A2 )
% 2.72/1.60 => ( ord_less_eq @ A @ ( minus_minus @ A @ C @ A2 ) @ ( minus_minus @ A @ C @ B ) ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_107_diff__mono,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( ordered_ab_group_add @ A )
% 2.72/1.60 => ! [A2: A,B: A,D: A,C: A] :
% 2.72/1.60 ( ( ord_less_eq @ A @ A2 @ B )
% 2.72/1.60 => ( ( ord_less_eq @ A @ D @ C )
% 2.72/1.60 => ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C ) @ ( minus_minus @ A @ B @ D ) ) ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_108_add__implies__diff,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( cancel1352612707id_add @ A )
% 2.72/1.60 => ! [C: A,B: A,A2: A] :
% 2.72/1.60 ( ( ( plus_plus @ A @ C @ B )
% 2.72/1.60 = A2 )
% 2.72/1.60 => ( C
% 2.72/1.60 = ( minus_minus @ A @ A2 @ B ) ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_109_diff__diff__add,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( cancel146912293up_add @ A )
% 2.72/1.60 => ! [A2: A,B: A,C: A] :
% 2.72/1.60 ( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B ) @ C )
% 2.72/1.60 = ( minus_minus @ A @ A2 @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_110_diff__add__eq__diff__diff__swap,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( group_add @ A )
% 2.72/1.60 => ! [A2: A,B: A,C: A] :
% 2.72/1.60 ( ( minus_minus @ A @ A2 @ ( plus_plus @ A @ B @ C ) )
% 2.72/1.60 = ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C ) @ B ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_111_diff__add__eq,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( ab_group_add @ A )
% 2.72/1.60 => ! [A2: A,B: A,C: A] :
% 2.72/1.60 ( ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B ) @ C )
% 2.72/1.60 = ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C ) @ B ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_112_diff__diff__eq2,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( group_add @ A )
% 2.72/1.60 => ! [A2: A,B: A,C: A] :
% 2.72/1.60 ( ( minus_minus @ A @ A2 @ ( minus_minus @ A @ B @ C ) )
% 2.72/1.60 = ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C ) @ B ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_113_add__diff__eq,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( group_add @ A )
% 2.72/1.60 => ! [A2: A,B: A,C: A] :
% 2.72/1.60 ( ( plus_plus @ A @ A2 @ ( minus_minus @ A @ B @ C ) )
% 2.72/1.60 = ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B ) @ C ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_114_eq__diff__eq,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( group_add @ A )
% 2.72/1.60 => ! [A2: A,C: A,B: A] :
% 2.72/1.60 ( ( A2
% 2.72/1.60 = ( minus_minus @ A @ C @ B ) )
% 2.72/1.60 = ( ( plus_plus @ A @ A2 @ B )
% 2.72/1.60 = C ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_115_diff__eq__eq,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( group_add @ A )
% 2.72/1.60 => ! [A2: A,B: A,C: A] :
% 2.72/1.60 ( ( ( minus_minus @ A @ A2 @ B )
% 2.72/1.60 = C )
% 2.72/1.60 = ( A2
% 2.72/1.60 = ( plus_plus @ A @ C @ B ) ) ) ) ).
% 2.72/1.60
% 2.72/1.60 thf(fact_116_group__cancel_Osub1,axiom,
% 2.72/1.60 ! [A: $tType] :
% 2.72/1.60 ( ( ab_group_add @ A )
% 2.72/1.60 => ! [A4: A,K: A,A2: A,B: A] :
% 2.72/1.60 ( ( A4
% 2.72/1.60 = ( plus_plus @ A @ K @ A2 ) )
% 2.72/1.60 => ( ( minus_minus @ A @ A4 @ B )
% 2.72/1.61 = ( plus_plus @ A @ K @ ( minus_minus @ A @ A2 @ B ) ) ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_117_abs__ge__self,axiom,
% 2.72/1.61 ! [A: $tType] :
% 2.72/1.61 ( ( ordere142940540dd_abs @ A )
% 2.72/1.61 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ ( abs_abs @ A @ A2 ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_118_abs__le__D1,axiom,
% 2.72/1.61 ! [A: $tType] :
% 2.72/1.61 ( ( ordere142940540dd_abs @ A )
% 2.72/1.61 => ! [A2: A,B: A] :
% 2.72/1.61 ( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ B )
% 2.72/1.61 => ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_119_abs__one,axiom,
% 2.72/1.61 ! [A: $tType] :
% 2.72/1.61 ( ( linordered_idom @ A )
% 2.72/1.61 => ( ( abs_abs @ A @ ( one_one @ A ) )
% 2.72/1.61 = ( one_one @ A ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_120_abs__minus__commute,axiom,
% 2.72/1.61 ! [A: $tType] :
% 2.72/1.61 ( ( ordere142940540dd_abs @ A )
% 2.72/1.61 => ! [A2: A,B: A] :
% 2.72/1.61 ( ( abs_abs @ A @ ( minus_minus @ A @ A2 @ B ) )
% 2.72/1.61 = ( abs_abs @ A @ ( minus_minus @ A @ B @ A2 ) ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_121_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
% 2.72/1.61 ! [A: $tType] :
% 2.72/1.61 ( ( ordere623563068d_diff @ A )
% 2.72/1.61 => ! [A2: A,B: A,C: A] :
% 2.72/1.61 ( ( ord_less_eq @ A @ A2 @ B )
% 2.72/1.61 => ( ( ord_less_eq @ A @ A2 @ B )
% 2.72/1.61 => ( ( ( minus_minus @ A @ B @ A2 )
% 2.72/1.61 = C )
% 2.72/1.61 = ( B
% 2.72/1.61 = ( plus_plus @ A @ C @ A2 ) ) ) ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_122_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
% 2.72/1.61 ! [A: $tType] :
% 2.72/1.61 ( ( ordere623563068d_diff @ A )
% 2.72/1.61 => ! [A2: A,B: A] :
% 2.72/1.61 ( ( ord_less_eq @ A @ A2 @ B )
% 2.72/1.61 => ( ( plus_plus @ A @ A2 @ ( minus_minus @ A @ B @ A2 ) )
% 2.72/1.61 = B ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_123_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
% 2.72/1.61 ! [A: $tType] :
% 2.72/1.61 ( ( ordere623563068d_diff @ A )
% 2.72/1.61 => ! [A2: A,B: A,C: A] :
% 2.72/1.61 ( ( ord_less_eq @ A @ A2 @ B )
% 2.72/1.61 => ( ( minus_minus @ A @ C @ ( minus_minus @ A @ B @ A2 ) )
% 2.72/1.61 = ( minus_minus @ A @ ( plus_plus @ A @ C @ A2 ) @ B ) ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_124_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
% 2.72/1.61 ! [A: $tType] :
% 2.72/1.61 ( ( ordere623563068d_diff @ A )
% 2.72/1.61 => ! [A2: A,B: A,C: A] :
% 2.72/1.61 ( ( ord_less_eq @ A @ A2 @ B )
% 2.72/1.61 => ( ( minus_minus @ A @ ( plus_plus @ A @ B @ C ) @ A2 )
% 2.72/1.61 = ( plus_plus @ A @ ( minus_minus @ A @ B @ A2 ) @ C ) ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_125_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
% 2.72/1.61 ! [A: $tType] :
% 2.72/1.61 ( ( ordere623563068d_diff @ A )
% 2.72/1.61 => ! [A2: A,B: A,C: A] :
% 2.72/1.61 ( ( ord_less_eq @ A @ A2 @ B )
% 2.72/1.61 => ( ( plus_plus @ A @ ( minus_minus @ A @ B @ A2 ) @ C )
% 2.72/1.61 = ( minus_minus @ A @ ( plus_plus @ A @ B @ C ) @ A2 ) ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_126_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
% 2.72/1.61 ! [A: $tType] :
% 2.72/1.61 ( ( ordere623563068d_diff @ A )
% 2.72/1.61 => ! [A2: A,B: A,C: A] :
% 2.72/1.61 ( ( ord_less_eq @ A @ A2 @ B )
% 2.72/1.61 => ( ( minus_minus @ A @ ( plus_plus @ A @ C @ B ) @ A2 )
% 2.72/1.61 = ( plus_plus @ A @ C @ ( minus_minus @ A @ B @ A2 ) ) ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_127_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
% 2.72/1.61 ! [A: $tType] :
% 2.72/1.61 ( ( ordere623563068d_diff @ A )
% 2.72/1.61 => ! [A2: A,B: A,C: A] :
% 2.72/1.61 ( ( ord_less_eq @ A @ A2 @ B )
% 2.72/1.61 => ( ( plus_plus @ A @ C @ ( minus_minus @ A @ B @ A2 ) )
% 2.72/1.61 = ( minus_minus @ A @ ( plus_plus @ A @ C @ B ) @ A2 ) ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_128_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
% 2.72/1.61 ! [A: $tType] :
% 2.72/1.61 ( ( ordere623563068d_diff @ A )
% 2.72/1.61 => ! [A2: A,B: A,C: A] :
% 2.72/1.61 ( ( ord_less_eq @ A @ A2 @ B )
% 2.72/1.61 => ( ( ord_less_eq @ A @ C @ ( minus_minus @ A @ B @ A2 ) )
% 2.72/1.61 = ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A2 ) @ B ) ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_129_le__add__diff,axiom,
% 2.72/1.61 ! [A: $tType] :
% 2.72/1.61 ( ( ordere623563068d_diff @ A )
% 2.72/1.61 => ! [A2: A,B: A,C: A] :
% 2.72/1.61 ( ( ord_less_eq @ A @ A2 @ B )
% 2.72/1.61 => ( ord_less_eq @ A @ C @ ( minus_minus @ A @ ( plus_plus @ A @ B @ C ) @ A2 ) ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_130_add__le__add__imp__diff__le,axiom,
% 2.72/1.61 ! [A: $tType] :
% 2.72/1.61 ( ( linordered_semidom @ A )
% 2.72/1.61 => ! [I2: A,K: A,N: A,J: A] :
% 2.72/1.61 ( ( ord_less_eq @ A @ ( plus_plus @ A @ I2 @ K ) @ N )
% 2.72/1.61 => ( ( ord_less_eq @ A @ N @ ( plus_plus @ A @ J @ K ) )
% 2.72/1.61 => ( ( ord_less_eq @ A @ ( plus_plus @ A @ I2 @ K ) @ N )
% 2.72/1.61 => ( ( ord_less_eq @ A @ N @ ( plus_plus @ A @ J @ K ) )
% 2.72/1.61 => ( ord_less_eq @ A @ ( minus_minus @ A @ N @ K ) @ J ) ) ) ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_131_diff__add,axiom,
% 2.72/1.61 ! [A: $tType] :
% 2.72/1.61 ( ( ordere623563068d_diff @ A )
% 2.72/1.61 => ! [A2: A,B: A] :
% 2.72/1.61 ( ( ord_less_eq @ A @ A2 @ B )
% 2.72/1.61 => ( ( plus_plus @ A @ ( minus_minus @ A @ B @ A2 ) @ A2 )
% 2.72/1.61 = B ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_132_add__le__imp__le__diff,axiom,
% 2.72/1.61 ! [A: $tType] :
% 2.72/1.61 ( ( linordered_semidom @ A )
% 2.72/1.61 => ! [I2: A,K: A,N: A] :
% 2.72/1.61 ( ( ord_less_eq @ A @ ( plus_plus @ A @ I2 @ K ) @ N )
% 2.72/1.61 => ( ord_less_eq @ A @ I2 @ ( minus_minus @ A @ N @ K ) ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_133_le__diff__eq,axiom,
% 2.72/1.61 ! [A: $tType] :
% 2.72/1.61 ( ( ordered_ab_group_add @ A )
% 2.72/1.61 => ! [A2: A,C: A,B: A] :
% 2.72/1.61 ( ( ord_less_eq @ A @ A2 @ ( minus_minus @ A @ C @ B ) )
% 2.72/1.61 = ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B ) @ C ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_134_diff__le__eq,axiom,
% 2.72/1.61 ! [A: $tType] :
% 2.72/1.61 ( ( ordered_ab_group_add @ A )
% 2.72/1.61 => ! [A2: A,B: A,C: A] :
% 2.72/1.61 ( ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ B ) @ C )
% 2.72/1.61 = ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ C @ B ) ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_135_abs__triangle__ineq,axiom,
% 2.72/1.61 ! [A: $tType] :
% 2.72/1.61 ( ( ordere142940540dd_abs @ A )
% 2.72/1.61 => ! [A2: A,B: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( plus_plus @ A @ A2 @ B ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B ) ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_136_abs__triangle__ineq2__sym,axiom,
% 2.72/1.61 ! [A: $tType] :
% 2.72/1.61 ( ( ordere142940540dd_abs @ A )
% 2.72/1.61 => ! [A2: A,B: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ B @ A2 ) ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_137_abs__triangle__ineq3,axiom,
% 2.72/1.61 ! [A: $tType] :
% 2.72/1.61 ( ( ordere142940540dd_abs @ A )
% 2.72/1.61 => ! [A2: A,B: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B ) ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ A2 @ B ) ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_138_abs__triangle__ineq2,axiom,
% 2.72/1.61 ! [A: $tType] :
% 2.72/1.61 ( ( ordere142940540dd_abs @ A )
% 2.72/1.61 => ! [A2: A,B: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ A2 @ B ) ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_139_abs__diff__triangle__ineq,axiom,
% 2.72/1.61 ! [A: $tType] :
% 2.72/1.61 ( ( ordere142940540dd_abs @ A )
% 2.72/1.61 => ! [A2: A,B: A,C: A,D: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B ) @ ( plus_plus @ A @ C @ D ) ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ ( minus_minus @ A @ A2 @ C ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ B @ D ) ) ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_140_abs__triangle__ineq4,axiom,
% 2.72/1.61 ! [A: $tType] :
% 2.72/1.61 ( ( ordere142940540dd_abs @ A )
% 2.72/1.61 => ! [A2: A,B: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ A2 @ B ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B ) ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_141_abs__diff__le__iff,axiom,
% 2.72/1.61 ! [A: $tType] :
% 2.72/1.61 ( ( linordered_idom @ A )
% 2.72/1.61 => ! [X: A,A2: A,R: A] :
% 2.72/1.61 ( ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X @ A2 ) ) @ R )
% 2.72/1.61 = ( ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ R ) @ X )
% 2.72/1.61 & ( ord_less_eq @ A @ X @ ( plus_plus @ A @ A2 @ R ) ) ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_142_posX,axiom,
% 2.72/1.61 ord_less_eq @ real @ ( zero_zero @ real ) @ ( minus_minus @ real @ ( genClo1161277105lle_PC @ l @ ( genClo1163638703lle_te @ p @ ( plus_plus @ nat @ i @ ( one_one @ nat ) ) ) ) @ ( genClo1161277105lle_PC @ l @ ( genClo1163638703lle_te @ q @ ( plus_plus @ nat @ i @ ( one_one @ nat ) ) ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_143_diff__numeral__special_I9_J,axiom,
% 2.72/1.61 ! [A: $tType] :
% 2.72/1.61 ( ( neg_numeral @ A )
% 2.72/1.61 => ( ( minus_minus @ A @ ( one_one @ A ) @ ( one_one @ A ) )
% 2.72/1.61 = ( zero_zero @ A ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_144_beta__bound2,axiom,
% 2.72/1.61 ! [P2: nat,I2: nat,Q2: nat] :
% 2.72/1.61 ( ( genClo1015804716orrect @ P2 @ ( genClo1163638703lle_te @ P2 @ ( plus_plus @ nat @ I2 @ ( one_one @ nat ) ) ) )
% 2.72/1.61 => ( ( genClo1015804716orrect @ Q2 @ ( genClo1163638703lle_te @ Q2 @ I2 ) )
% 2.72/1.61 => ( ord_less_eq @ real @ ( minus_minus @ real @ ( genClo1163638703lle_te @ P2 @ ( plus_plus @ nat @ I2 @ ( one_one @ nat ) ) ) @ ( genClo1163638703lle_te @ Q2 @ I2 ) ) @ ( plus_plus @ real @ genClo1650508560e_rmax @ genClo1278781456e_beta ) ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_145_rts1d,axiom,
% 2.72/1.61 ! [P2: nat,I2: nat] :
% 2.72/1.61 ( ( genClo1015804716orrect @ P2 @ ( genClo1163638703lle_te @ P2 @ ( plus_plus @ nat @ I2 @ ( one_one @ nat ) ) ) )
% 2.72/1.61 => ( ord_less_eq @ real @ genClo1651033342e_rmin @ ( minus_minus @ real @ ( genClo1163638703lle_te @ P2 @ ( plus_plus @ nat @ I2 @ ( one_one @ nat ) ) ) @ ( genClo1163638703lle_te @ P2 @ I2 ) ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_146_rts1,axiom,
% 2.72/1.61 ! [P: nat,T: real,I: nat] :
% 2.72/1.61 ( ( ( genClo1015804716orrect @ P @ T )
% 2.72/1.61 & ( ord_less_eq @ real @ ( genClo1163638703lle_te @ P @ ( plus_plus @ nat @ I @ ( one_one @ nat ) ) ) @ T ) )
% 2.72/1.61 => ( ord_less_eq @ real @ genClo1651033342e_rmin @ ( minus_minus @ real @ T @ ( genClo1163638703lle_te @ P @ I ) ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_147_rts0d,axiom,
% 2.72/1.61 ! [P2: nat,I2: nat] :
% 2.72/1.61 ( ( genClo1015804716orrect @ P2 @ ( genClo1163638703lle_te @ P2 @ ( plus_plus @ nat @ I2 @ ( one_one @ nat ) ) ) )
% 2.72/1.61 => ( ord_less_eq @ real @ ( minus_minus @ real @ ( genClo1163638703lle_te @ P2 @ ( plus_plus @ nat @ I2 @ ( one_one @ nat ) ) ) @ ( genClo1163638703lle_te @ P2 @ I2 ) ) @ genClo1650508560e_rmax ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_148_rts0,axiom,
% 2.72/1.61 ! [P: nat,T: real,I: nat] :
% 2.72/1.61 ( ( ( genClo1015804716orrect @ P @ T )
% 2.72/1.61 & ( ord_less_eq @ real @ T @ ( genClo1163638703lle_te @ P @ ( plus_plus @ nat @ I @ ( one_one @ nat ) ) ) ) )
% 2.72/1.61 => ( ord_less_eq @ real @ ( minus_minus @ real @ T @ ( genClo1163638703lle_te @ P @ I ) ) @ genClo1650508560e_rmax ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_149_okRead2__def,axiom,
% 2.72/1.61 ( genClo293725282kRead2
% 2.72/1.61 = ( ^ [F2: nat > real,G2: nat > real,X3: real,Ppred: nat > $o] :
% 2.72/1.61 ! [P3: nat] :
% 2.72/1.61 ( ( Ppred @ P3 )
% 2.72/1.61 => ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( F2 @ P3 ) @ ( G2 @ P3 ) ) ) @ X3 ) ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_150_sin__bound__lemma,axiom,
% 2.72/1.61 ! [X: real,Y: real,U: real,V: real] :
% 2.72/1.61 ( ( X = Y )
% 2.72/1.61 => ( ( ord_less_eq @ real @ ( abs_abs @ real @ U ) @ V )
% 2.72/1.61 => ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( plus_plus @ real @ X @ U ) @ Y ) ) @ V ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_151_Nat_Oadd__diff__assoc,axiom,
% 2.72/1.61 ! [K: nat,J: nat,I2: nat] :
% 2.72/1.61 ( ( ord_less_eq @ nat @ K @ J )
% 2.72/1.61 => ( ( plus_plus @ nat @ I2 @ ( minus_minus @ nat @ J @ K ) )
% 2.72/1.61 = ( minus_minus @ nat @ ( plus_plus @ nat @ I2 @ J ) @ K ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_152_le0,axiom,
% 2.72/1.61 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_153_bot__nat__0_Oextremum,axiom,
% 2.72/1.61 ! [A2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A2 ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_154_add__is__0,axiom,
% 2.72/1.61 ! [M: nat,N: nat] :
% 2.72/1.61 ( ( ( plus_plus @ nat @ M @ N )
% 2.72/1.61 = ( zero_zero @ nat ) )
% 2.72/1.61 = ( ( M
% 2.72/1.61 = ( zero_zero @ nat ) )
% 2.72/1.61 & ( N
% 2.72/1.61 = ( zero_zero @ nat ) ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_155_Nat_Oadd__0__right,axiom,
% 2.72/1.61 ! [M: nat] :
% 2.72/1.61 ( ( plus_plus @ nat @ M @ ( zero_zero @ nat ) )
% 2.72/1.61 = M ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_156_diff__0__eq__0,axiom,
% 2.72/1.61 ! [N: nat] :
% 2.72/1.61 ( ( minus_minus @ nat @ ( zero_zero @ nat ) @ N )
% 2.72/1.61 = ( zero_zero @ nat ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_157_diff__self__eq__0,axiom,
% 2.72/1.61 ! [M: nat] :
% 2.72/1.61 ( ( minus_minus @ nat @ M @ M )
% 2.72/1.61 = ( zero_zero @ nat ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_158_nat__add__left__cancel__le,axiom,
% 2.72/1.61 ! [K: nat,M: nat,N: nat] :
% 2.72/1.61 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N ) )
% 2.72/1.61 = ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_159_diff__is__0__eq,axiom,
% 2.72/1.61 ! [M: nat,N: nat] :
% 2.72/1.61 ( ( ( minus_minus @ nat @ M @ N )
% 2.72/1.61 = ( zero_zero @ nat ) )
% 2.72/1.61 = ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_160_diff__is__0__eq_H,axiom,
% 2.72/1.61 ! [M: nat,N: nat] :
% 2.72/1.61 ( ( ord_less_eq @ nat @ M @ N )
% 2.72/1.61 => ( ( minus_minus @ nat @ M @ N )
% 2.72/1.61 = ( zero_zero @ nat ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_161_diff__diff__cancel,axiom,
% 2.72/1.61 ! [I2: nat,N: nat] :
% 2.72/1.61 ( ( ord_less_eq @ nat @ I2 @ N )
% 2.72/1.61 => ( ( minus_minus @ nat @ N @ ( minus_minus @ nat @ N @ I2 ) )
% 2.72/1.61 = I2 ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_162_diff__diff__left,axiom,
% 2.72/1.61 ! [I2: nat,J: nat,K: nat] :
% 2.72/1.61 ( ( minus_minus @ nat @ ( minus_minus @ nat @ I2 @ J ) @ K )
% 2.72/1.61 = ( minus_minus @ nat @ I2 @ ( plus_plus @ nat @ J @ K ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_163_Nat_Odiff__diff__right,axiom,
% 2.72/1.61 ! [K: nat,J: nat,I2: nat] :
% 2.72/1.61 ( ( ord_less_eq @ nat @ K @ J )
% 2.72/1.61 => ( ( minus_minus @ nat @ I2 @ ( minus_minus @ nat @ J @ K ) )
% 2.72/1.61 = ( minus_minus @ nat @ ( plus_plus @ nat @ I2 @ K ) @ J ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_164_Nat_Oadd__diff__assoc2,axiom,
% 2.72/1.61 ! [K: nat,J: nat,I2: nat] :
% 2.72/1.61 ( ( ord_less_eq @ nat @ K @ J )
% 2.72/1.61 => ( ( plus_plus @ nat @ ( minus_minus @ nat @ J @ K ) @ I2 )
% 2.72/1.61 = ( minus_minus @ nat @ ( plus_plus @ nat @ J @ I2 ) @ K ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_165_less__eq__nat_Osimps_I1_J,axiom,
% 2.72/1.61 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_166_le__0__eq,axiom,
% 2.72/1.61 ! [N: nat] :
% 2.72/1.61 ( ( ord_less_eq @ nat @ N @ ( zero_zero @ nat ) )
% 2.72/1.61 = ( N
% 2.72/1.61 = ( zero_zero @ nat ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_167_le__refl,axiom,
% 2.72/1.61 ! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_168_le__trans,axiom,
% 2.72/1.61 ! [I2: nat,J: nat,K: nat] :
% 2.72/1.61 ( ( ord_less_eq @ nat @ I2 @ J )
% 2.72/1.61 => ( ( ord_less_eq @ nat @ J @ K )
% 2.72/1.61 => ( ord_less_eq @ nat @ I2 @ K ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_169_eq__imp__le,axiom,
% 2.72/1.61 ! [M: nat,N: nat] :
% 2.72/1.61 ( ( M = N )
% 2.72/1.61 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_170_le__antisym,axiom,
% 2.72/1.61 ! [M: nat,N: nat] :
% 2.72/1.61 ( ( ord_less_eq @ nat @ M @ N )
% 2.72/1.61 => ( ( ord_less_eq @ nat @ N @ M )
% 2.72/1.61 => ( M = N ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_171_eq__diff__iff,axiom,
% 2.72/1.61 ! [K: nat,M: nat,N: nat] :
% 2.72/1.61 ( ( ord_less_eq @ nat @ K @ M )
% 2.72/1.61 => ( ( ord_less_eq @ nat @ K @ N )
% 2.72/1.61 => ( ( ( minus_minus @ nat @ M @ K )
% 2.72/1.61 = ( minus_minus @ nat @ N @ K ) )
% 2.72/1.61 = ( M = N ) ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_172_le__diff__iff,axiom,
% 2.72/1.61 ! [K: nat,M: nat,N: nat] :
% 2.72/1.61 ( ( ord_less_eq @ nat @ K @ M )
% 2.72/1.61 => ( ( ord_less_eq @ nat @ K @ N )
% 2.72/1.61 => ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
% 2.72/1.61 = ( ord_less_eq @ nat @ M @ N ) ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_173_diff__commute,axiom,
% 2.72/1.61 ! [I2: nat,J: nat,K: nat] :
% 2.72/1.61 ( ( minus_minus @ nat @ ( minus_minus @ nat @ I2 @ J ) @ K )
% 2.72/1.61 = ( minus_minus @ nat @ ( minus_minus @ nat @ I2 @ K ) @ J ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_174_Nat_Odiff__diff__eq,axiom,
% 2.72/1.61 ! [K: nat,M: nat,N: nat] :
% 2.72/1.61 ( ( ord_less_eq @ nat @ K @ M )
% 2.72/1.61 => ( ( ord_less_eq @ nat @ K @ N )
% 2.72/1.61 => ( ( minus_minus @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
% 2.72/1.61 = ( minus_minus @ nat @ M @ N ) ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_175_diff__le__mono,axiom,
% 2.72/1.61 ! [M: nat,N: nat,L: nat] :
% 2.72/1.61 ( ( ord_less_eq @ nat @ M @ N )
% 2.72/1.61 => ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ L ) @ ( minus_minus @ nat @ N @ L ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_176_diff__le__self,axiom,
% 2.72/1.61 ! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ N ) @ M ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_177_le__diff__iff_H,axiom,
% 2.72/1.61 ! [A2: nat,C: nat,B: nat] :
% 2.72/1.61 ( ( ord_less_eq @ nat @ A2 @ C )
% 2.72/1.61 => ( ( ord_less_eq @ nat @ B @ C )
% 2.72/1.61 => ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ C @ A2 ) @ ( minus_minus @ nat @ C @ B ) )
% 2.72/1.61 = ( ord_less_eq @ nat @ B @ A2 ) ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_178_diff__le__mono2,axiom,
% 2.72/1.61 ! [M: nat,N: nat,L: nat] :
% 2.72/1.61 ( ( ord_less_eq @ nat @ M @ N )
% 2.72/1.61 => ( ord_less_eq @ nat @ ( minus_minus @ nat @ L @ N ) @ ( minus_minus @ nat @ L @ M ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_179_nat__le__linear,axiom,
% 2.72/1.61 ! [M: nat,N: nat] :
% 2.72/1.61 ( ( ord_less_eq @ nat @ M @ N )
% 2.72/1.61 | ( ord_less_eq @ nat @ N @ M ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_180_diffs0__imp__equal,axiom,
% 2.72/1.61 ! [M: nat,N: nat] :
% 2.72/1.61 ( ( ( minus_minus @ nat @ M @ N )
% 2.72/1.61 = ( zero_zero @ nat ) )
% 2.72/1.61 => ( ( ( minus_minus @ nat @ N @ M )
% 2.72/1.61 = ( zero_zero @ nat ) )
% 2.72/1.61 => ( M = N ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_181_minus__nat_Odiff__0,axiom,
% 2.72/1.61 ! [M: nat] :
% 2.72/1.61 ( ( minus_minus @ nat @ M @ ( zero_zero @ nat ) )
% 2.72/1.61 = M ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_182_Nat_Oex__has__greatest__nat,axiom,
% 2.72/1.61 ! [P4: nat > $o,K: nat,B: nat] :
% 2.72/1.61 ( ( P4 @ K )
% 2.72/1.61 => ( ! [Y3: nat] :
% 2.72/1.61 ( ( P4 @ Y3 )
% 2.72/1.61 => ( ord_less_eq @ nat @ Y3 @ B ) )
% 2.72/1.61 => ? [X2: nat] :
% 2.72/1.61 ( ( P4 @ X2 )
% 2.72/1.61 & ! [Y4: nat] :
% 2.72/1.61 ( ( P4 @ Y4 )
% 2.72/1.61 => ( ord_less_eq @ nat @ Y4 @ X2 ) ) ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_183_bot__nat__0_Oextremum__unique,axiom,
% 2.72/1.61 ! [A2: nat] :
% 2.72/1.61 ( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
% 2.72/1.61 = ( A2
% 2.72/1.61 = ( zero_zero @ nat ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_184_bot__nat__0_Oextremum__uniqueI,axiom,
% 2.72/1.61 ! [A2: nat] :
% 2.72/1.61 ( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
% 2.72/1.61 => ( A2
% 2.72/1.61 = ( zero_zero @ nat ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_185_plus__nat_Oadd__0,axiom,
% 2.72/1.61 ! [N: nat] :
% 2.72/1.61 ( ( plus_plus @ nat @ ( zero_zero @ nat ) @ N )
% 2.72/1.61 = N ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_186_add__eq__self__zero,axiom,
% 2.72/1.61 ! [M: nat,N: nat] :
% 2.72/1.61 ( ( ( plus_plus @ nat @ M @ N )
% 2.72/1.61 = M )
% 2.72/1.61 => ( N
% 2.72/1.61 = ( zero_zero @ nat ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_187_diff__add__0,axiom,
% 2.72/1.61 ! [N: nat,M: nat] :
% 2.72/1.61 ( ( minus_minus @ nat @ N @ ( plus_plus @ nat @ N @ M ) )
% 2.72/1.61 = ( zero_zero @ nat ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_188_synch0,axiom,
% 2.72/1.61 ! [P: nat] :
% 2.72/1.61 ( ( genClo1163638703lle_te @ P @ ( zero_zero @ nat ) )
% 2.72/1.61 = ( zero_zero @ real ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_189_is__num__normalize_I1_J,axiom,
% 2.72/1.61 ! [A: $tType] :
% 2.72/1.61 ( ( neg_numeral @ A )
% 2.72/1.61 => ! [A2: A,B: A,C: A] :
% 2.72/1.61 ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B ) @ C )
% 2.72/1.61 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_190_add__leE,axiom,
% 2.72/1.61 ! [M: nat,K: nat,N: nat] :
% 2.72/1.61 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K ) @ N )
% 2.72/1.61 => ~ ( ( ord_less_eq @ nat @ M @ N )
% 2.72/1.61 => ~ ( ord_less_eq @ nat @ K @ N ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_191_le__add1,axiom,
% 2.72/1.61 ! [N: nat,M: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ N @ M ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_192_le__add2,axiom,
% 2.72/1.61 ! [N: nat,M: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ M @ N ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_193_add__leD1,axiom,
% 2.72/1.61 ! [M: nat,K: nat,N: nat] :
% 2.72/1.61 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K ) @ N )
% 2.72/1.61 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_194_add__leD2,axiom,
% 2.72/1.61 ! [M: nat,K: nat,N: nat] :
% 2.72/1.61 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K ) @ N )
% 2.72/1.61 => ( ord_less_eq @ nat @ K @ N ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_195_le__Suc__ex,axiom,
% 2.72/1.61 ! [K: nat,L: nat] :
% 2.72/1.61 ( ( ord_less_eq @ nat @ K @ L )
% 2.72/1.61 => ? [N2: nat] :
% 2.72/1.61 ( L
% 2.72/1.61 = ( plus_plus @ nat @ K @ N2 ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_196_add__le__mono,axiom,
% 2.72/1.61 ! [I2: nat,J: nat,K: nat,L: nat] :
% 2.72/1.61 ( ( ord_less_eq @ nat @ I2 @ J )
% 2.72/1.61 => ( ( ord_less_eq @ nat @ K @ L )
% 2.72/1.61 => ( ord_less_eq @ nat @ ( plus_plus @ nat @ I2 @ K ) @ ( plus_plus @ nat @ J @ L ) ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_197_add__le__mono1,axiom,
% 2.72/1.61 ! [I2: nat,J: nat,K: nat] :
% 2.72/1.61 ( ( ord_less_eq @ nat @ I2 @ J )
% 2.72/1.61 => ( ord_less_eq @ nat @ ( plus_plus @ nat @ I2 @ K ) @ ( plus_plus @ nat @ J @ K ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_198_trans__le__add1,axiom,
% 2.72/1.61 ! [I2: nat,J: nat,M: nat] :
% 2.72/1.61 ( ( ord_less_eq @ nat @ I2 @ J )
% 2.72/1.61 => ( ord_less_eq @ nat @ I2 @ ( plus_plus @ nat @ J @ M ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_199_trans__le__add2,axiom,
% 2.72/1.61 ! [I2: nat,J: nat,M: nat] :
% 2.72/1.61 ( ( ord_less_eq @ nat @ I2 @ J )
% 2.72/1.61 => ( ord_less_eq @ nat @ I2 @ ( plus_plus @ nat @ M @ J ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_200_nat__le__iff__add,axiom,
% 2.72/1.61 ( ( ord_less_eq @ nat )
% 2.72/1.61 = ( ^ [M2: nat,N3: nat] :
% 2.72/1.61 ? [K2: nat] :
% 2.72/1.61 ( N3
% 2.72/1.61 = ( plus_plus @ nat @ M2 @ K2 ) ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_201_Nat_Odiff__cancel,axiom,
% 2.72/1.61 ! [K: nat,M: nat,N: nat] :
% 2.72/1.61 ( ( minus_minus @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N ) )
% 2.72/1.61 = ( minus_minus @ nat @ M @ N ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_202_diff__cancel2,axiom,
% 2.72/1.61 ! [M: nat,K: nat,N: nat] :
% 2.72/1.61 ( ( minus_minus @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N @ K ) )
% 2.72/1.61 = ( minus_minus @ nat @ M @ N ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_203_diff__add__inverse,axiom,
% 2.72/1.61 ! [N: nat,M: nat] :
% 2.72/1.61 ( ( minus_minus @ nat @ ( plus_plus @ nat @ N @ M ) @ N )
% 2.72/1.61 = M ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_204_diff__add__inverse2,axiom,
% 2.72/1.61 ! [M: nat,N: nat] :
% 2.72/1.61 ( ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ N )
% 2.72/1.61 = M ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_205_Nat_Ole__imp__diff__is__add,axiom,
% 2.72/1.61 ! [I2: nat,J: nat,K: nat] :
% 2.72/1.61 ( ( ord_less_eq @ nat @ I2 @ J )
% 2.72/1.61 => ( ( ( minus_minus @ nat @ J @ I2 )
% 2.72/1.61 = K )
% 2.72/1.61 = ( J
% 2.72/1.61 = ( plus_plus @ nat @ K @ I2 ) ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_206_Nat_Odiff__add__assoc2,axiom,
% 2.72/1.61 ! [K: nat,J: nat,I2: nat] :
% 2.72/1.61 ( ( ord_less_eq @ nat @ K @ J )
% 2.72/1.61 => ( ( minus_minus @ nat @ ( plus_plus @ nat @ J @ I2 ) @ K )
% 2.72/1.61 = ( plus_plus @ nat @ ( minus_minus @ nat @ J @ K ) @ I2 ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_207_Nat_Odiff__add__assoc,axiom,
% 2.72/1.61 ! [K: nat,J: nat,I2: nat] :
% 2.72/1.61 ( ( ord_less_eq @ nat @ K @ J )
% 2.72/1.61 => ( ( minus_minus @ nat @ ( plus_plus @ nat @ I2 @ J ) @ K )
% 2.72/1.61 = ( plus_plus @ nat @ I2 @ ( minus_minus @ nat @ J @ K ) ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_208_Nat_Ole__diff__conv2,axiom,
% 2.72/1.61 ! [K: nat,J: nat,I2: nat] :
% 2.72/1.61 ( ( ord_less_eq @ nat @ K @ J )
% 2.72/1.61 => ( ( ord_less_eq @ nat @ I2 @ ( minus_minus @ nat @ J @ K ) )
% 2.72/1.61 = ( ord_less_eq @ nat @ ( plus_plus @ nat @ I2 @ K ) @ J ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_209_le__diff__conv,axiom,
% 2.72/1.61 ! [J: nat,K: nat,I2: nat] :
% 2.72/1.61 ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ J @ K ) @ I2 )
% 2.72/1.61 = ( ord_less_eq @ nat @ J @ ( plus_plus @ nat @ I2 @ K ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_210_PC__monotone,axiom,
% 2.72/1.61 ! [P: nat,S: real,T: real] :
% 2.72/1.61 ( ( ( genClo1015804716orrect @ P @ T )
% 2.72/1.61 & ( ord_less_eq @ real @ S @ T ) )
% 2.72/1.61 => ( ord_less_eq @ real @ ( genClo1161277105lle_PC @ P @ S ) @ ( genClo1161277105lle_PC @ P @ T ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_211_eq__diff__eq_H,axiom,
% 2.72/1.61 ! [X: real,Y: real,Z2: real] :
% 2.72/1.61 ( ( X
% 2.72/1.61 = ( minus_minus @ real @ Y @ Z2 ) )
% 2.72/1.61 = ( Y
% 2.72/1.61 = ( plus_plus @ real @ X @ Z2 ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_212_le__numeral__extra_I3_J,axiom,
% 2.72/1.61 ! [A: $tType] :
% 2.72/1.61 ( ( linord1659791738miring @ A )
% 2.72/1.61 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_213_nonoverlap,axiom,
% 2.72/1.61 ord_less_eq @ real @ genClo1278781456e_beta @ genClo1651033342e_rmin ).
% 2.72/1.61
% 2.72/1.61 thf(fact_214_le__numeral__extra_I4_J,axiom,
% 2.72/1.61 ! [A: $tType] :
% 2.72/1.61 ( ( linord1659791738miring @ A )
% 2.72/1.61 => ( ord_less_eq @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_215_init,axiom,
% 2.72/1.61 ! [P: nat] :
% 2.72/1.61 ( ( genClo1015804716orrect @ P @ ( zero_zero @ real ) )
% 2.72/1.61 => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( genClo1161277105lle_PC @ P @ ( zero_zero @ real ) ) )
% 2.72/1.61 & ( ord_less_eq @ real @ ( genClo1161277105lle_PC @ P @ ( zero_zero @ real ) ) @ genClo872980712lle_mu ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_216_bound1,axiom,
% 2.72/1.61 ord_less_eq @ real @ ( minus_minus @ real @ ( genClo1161277105lle_PC @ l @ ( genClo1163638703lle_te @ p @ ( plus_plus @ nat @ i @ ( one_one @ nat ) ) ) ) @ ( genClo1161277105lle_PC @ l @ ( genClo1163638703lle_te @ q @ ( plus_plus @ nat @ i @ ( one_one @ nat ) ) ) ) ) @ ( times_times @ real @ ( minus_minus @ real @ ( genClo1163638703lle_te @ p @ ( plus_plus @ nat @ i @ ( one_one @ nat ) ) ) @ ( genClo1163638703lle_te @ q @ ( plus_plus @ nat @ i @ ( one_one @ nat ) ) ) ) @ ( plus_plus @ real @ ( one_one @ real ) @ genClo1144207539le_rho ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_217_minus__apply,axiom,
% 2.72/1.61 ! [B2: $tType,A: $tType] :
% 2.72/1.61 ( ( minus @ B2 )
% 2.72/1.61 => ( ( minus_minus @ ( A > B2 ) )
% 2.72/1.61 = ( ^ [A5: A > B2,B5: A > B2,X3: A] : ( minus_minus @ B2 @ ( A5 @ X3 ) @ ( B5 @ X3 ) ) ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_218_mult__cancel__right,axiom,
% 2.72/1.61 ! [A: $tType] :
% 2.72/1.61 ( ( semiri1923998003cancel @ A )
% 2.72/1.61 => ! [A2: A,C: A,B: A] :
% 2.72/1.61 ( ( ( times_times @ A @ A2 @ C )
% 2.72/1.61 = ( times_times @ A @ B @ C ) )
% 2.72/1.61 = ( ( C
% 2.72/1.61 = ( zero_zero @ A ) )
% 2.72/1.61 | ( A2 = B ) ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_219_mult__cancel__left,axiom,
% 2.72/1.61 ! [A: $tType] :
% 2.72/1.61 ( ( semiri1923998003cancel @ A )
% 2.72/1.61 => ! [C: A,A2: A,B: A] :
% 2.72/1.61 ( ( ( times_times @ A @ C @ A2 )
% 2.72/1.61 = ( times_times @ A @ C @ B ) )
% 2.72/1.61 = ( ( C
% 2.72/1.61 = ( zero_zero @ A ) )
% 2.72/1.61 | ( A2 = B ) ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_220_mult__eq__0__iff,axiom,
% 2.72/1.61 ! [A: $tType] :
% 2.72/1.61 ( ( semiri1193490041visors @ A )
% 2.72/1.61 => ! [A2: A,B: A] :
% 2.72/1.61 ( ( ( times_times @ A @ A2 @ B )
% 2.72/1.61 = ( zero_zero @ A ) )
% 2.72/1.61 = ( ( A2
% 2.72/1.61 = ( zero_zero @ A ) )
% 2.72/1.61 | ( B
% 2.72/1.61 = ( zero_zero @ A ) ) ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_221_mult__zero__right,axiom,
% 2.72/1.61 ! [A: $tType] :
% 2.72/1.61 ( ( mult_zero @ A )
% 2.72/1.61 => ! [A2: A] :
% 2.72/1.61 ( ( times_times @ A @ A2 @ ( zero_zero @ A ) )
% 2.72/1.61 = ( zero_zero @ A ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_222_mult__zero__left,axiom,
% 2.72/1.61 ! [A: $tType] :
% 2.72/1.61 ( ( mult_zero @ A )
% 2.72/1.61 => ! [A2: A] :
% 2.72/1.61 ( ( times_times @ A @ ( zero_zero @ A ) @ A2 )
% 2.72/1.61 = ( zero_zero @ A ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_223_mult_Oleft__neutral,axiom,
% 2.72/1.61 ! [A: $tType] :
% 2.72/1.61 ( ( monoid_mult @ A )
% 2.72/1.61 => ! [A2: A] :
% 2.72/1.61 ( ( times_times @ A @ ( one_one @ A ) @ A2 )
% 2.72/1.61 = A2 ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_224_mult_Oright__neutral,axiom,
% 2.72/1.61 ! [A: $tType] :
% 2.72/1.61 ( ( monoid_mult @ A )
% 2.72/1.61 => ! [A2: A] :
% 2.72/1.61 ( ( times_times @ A @ A2 @ ( one_one @ A ) )
% 2.72/1.61 = A2 ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_225_abs__mult__self__eq,axiom,
% 2.72/1.61 ! [A: $tType] :
% 2.72/1.61 ( ( linordered_idom @ A )
% 2.72/1.61 => ! [A2: A] :
% 2.72/1.61 ( ( times_times @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ A2 ) )
% 2.72/1.61 = ( times_times @ A @ A2 @ A2 ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_226_mult__cancel__left1,axiom,
% 2.72/1.61 ! [A: $tType] :
% 2.72/1.61 ( ( ring_11004092258visors @ A )
% 2.72/1.61 => ! [C: A,B: A] :
% 2.72/1.61 ( ( C
% 2.72/1.61 = ( times_times @ A @ C @ B ) )
% 2.72/1.61 = ( ( C
% 2.72/1.61 = ( zero_zero @ A ) )
% 2.72/1.61 | ( B
% 2.72/1.61 = ( one_one @ A ) ) ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_227_mult__cancel__left2,axiom,
% 2.72/1.61 ! [A: $tType] :
% 2.72/1.61 ( ( ring_11004092258visors @ A )
% 2.72/1.61 => ! [C: A,A2: A] :
% 2.72/1.61 ( ( ( times_times @ A @ C @ A2 )
% 2.72/1.61 = C )
% 2.72/1.61 = ( ( C
% 2.72/1.61 = ( zero_zero @ A ) )
% 2.72/1.61 | ( A2
% 2.72/1.61 = ( one_one @ A ) ) ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_228_mult__cancel__right1,axiom,
% 2.72/1.61 ! [A: $tType] :
% 2.72/1.61 ( ( ring_11004092258visors @ A )
% 2.72/1.61 => ! [C: A,B: A] :
% 2.72/1.61 ( ( C
% 2.72/1.61 = ( times_times @ A @ B @ C ) )
% 2.72/1.61 = ( ( C
% 2.72/1.61 = ( zero_zero @ A ) )
% 2.72/1.61 | ( B
% 2.72/1.61 = ( one_one @ A ) ) ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_229_mult__cancel__right2,axiom,
% 2.72/1.61 ! [A: $tType] :
% 2.72/1.61 ( ( ring_11004092258visors @ A )
% 2.72/1.61 => ! [A2: A,C: A] :
% 2.72/1.61 ( ( ( times_times @ A @ A2 @ C )
% 2.72/1.61 = C )
% 2.72/1.61 = ( ( C
% 2.72/1.61 = ( zero_zero @ A ) )
% 2.72/1.61 | ( A2
% 2.72/1.61 = ( one_one @ A ) ) ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_230_mult__right__cancel,axiom,
% 2.72/1.61 ! [A: $tType] :
% 2.72/1.61 ( ( semiri1923998003cancel @ A )
% 2.72/1.61 => ! [C: A,A2: A,B: A] :
% 2.72/1.61 ( ( C
% 2.72/1.61 != ( zero_zero @ A ) )
% 2.72/1.61 => ( ( ( times_times @ A @ A2 @ C )
% 2.72/1.61 = ( times_times @ A @ B @ C ) )
% 2.72/1.61 = ( A2 = B ) ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_231_mult__left__cancel,axiom,
% 2.72/1.61 ! [A: $tType] :
% 2.72/1.61 ( ( semiri1923998003cancel @ A )
% 2.72/1.61 => ! [C: A,A2: A,B: A] :
% 2.72/1.61 ( ( C
% 2.72/1.61 != ( zero_zero @ A ) )
% 2.72/1.61 => ( ( ( times_times @ A @ C @ A2 )
% 2.72/1.61 = ( times_times @ A @ C @ B ) )
% 2.72/1.61 = ( A2 = B ) ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_232_no__zero__divisors,axiom,
% 2.72/1.61 ! [A: $tType] :
% 2.72/1.61 ( ( semiri1193490041visors @ A )
% 2.72/1.61 => ! [A2: A,B: A] :
% 2.72/1.61 ( ( A2
% 2.72/1.61 != ( zero_zero @ A ) )
% 2.72/1.61 => ( ( B
% 2.72/1.61 != ( zero_zero @ A ) )
% 2.72/1.61 => ( ( times_times @ A @ A2 @ B )
% 2.72/1.61 != ( zero_zero @ A ) ) ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_233_divisors__zero,axiom,
% 2.72/1.61 ! [A: $tType] :
% 2.72/1.61 ( ( semiri1193490041visors @ A )
% 2.72/1.61 => ! [A2: A,B: A] :
% 2.72/1.61 ( ( ( times_times @ A @ A2 @ B )
% 2.72/1.61 = ( zero_zero @ A ) )
% 2.72/1.61 => ( ( A2
% 2.72/1.61 = ( zero_zero @ A ) )
% 2.72/1.61 | ( B
% 2.72/1.61 = ( zero_zero @ A ) ) ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_234_mult__not__zero,axiom,
% 2.72/1.61 ! [A: $tType] :
% 2.72/1.61 ( ( mult_zero @ A )
% 2.72/1.61 => ! [A2: A,B: A] :
% 2.72/1.61 ( ( ( times_times @ A @ A2 @ B )
% 2.72/1.61 != ( zero_zero @ A ) )
% 2.72/1.61 => ( ( A2
% 2.72/1.61 != ( zero_zero @ A ) )
% 2.72/1.61 & ( B
% 2.72/1.61 != ( zero_zero @ A ) ) ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_235_ring__class_Oring__distribs_I2_J,axiom,
% 2.72/1.61 ! [A: $tType] :
% 2.72/1.61 ( ( ring @ A )
% 2.72/1.61 => ! [A2: A,B: A,C: A] :
% 2.72/1.61 ( ( times_times @ A @ ( plus_plus @ A @ A2 @ B ) @ C )
% 2.72/1.61 = ( plus_plus @ A @ ( times_times @ A @ A2 @ C ) @ ( times_times @ A @ B @ C ) ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_236_ring__class_Oring__distribs_I1_J,axiom,
% 2.72/1.61 ! [A: $tType] :
% 2.72/1.61 ( ( ring @ A )
% 2.72/1.61 => ! [A2: A,B: A,C: A] :
% 2.72/1.61 ( ( times_times @ A @ A2 @ ( plus_plus @ A @ B @ C ) )
% 2.72/1.61 = ( plus_plus @ A @ ( times_times @ A @ A2 @ B ) @ ( times_times @ A @ A2 @ C ) ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_237_comm__semiring__class_Odistrib,axiom,
% 2.72/1.61 ! [A: $tType] :
% 2.72/1.61 ( ( comm_semiring @ A )
% 2.72/1.61 => ! [A2: A,B: A,C: A] :
% 2.72/1.61 ( ( times_times @ A @ ( plus_plus @ A @ A2 @ B ) @ C )
% 2.72/1.61 = ( plus_plus @ A @ ( times_times @ A @ A2 @ C ) @ ( times_times @ A @ B @ C ) ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_238_distrib__left,axiom,
% 2.72/1.61 ! [A: $tType] :
% 2.72/1.61 ( ( semiring @ A )
% 2.72/1.61 => ! [A2: A,B: A,C: A] :
% 2.72/1.61 ( ( times_times @ A @ A2 @ ( plus_plus @ A @ B @ C ) )
% 2.72/1.61 = ( plus_plus @ A @ ( times_times @ A @ A2 @ B ) @ ( times_times @ A @ A2 @ C ) ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_239_distrib__right,axiom,
% 2.72/1.61 ! [A: $tType] :
% 2.72/1.61 ( ( semiring @ A )
% 2.72/1.61 => ! [A2: A,B: A,C: A] :
% 2.72/1.61 ( ( times_times @ A @ ( plus_plus @ A @ A2 @ B ) @ C )
% 2.72/1.61 = ( plus_plus @ A @ ( times_times @ A @ A2 @ C ) @ ( times_times @ A @ B @ C ) ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_240_combine__common__factor,axiom,
% 2.72/1.61 ! [A: $tType] :
% 2.72/1.61 ( ( semiring @ A )
% 2.72/1.61 => ! [A2: A,E: A,B: A,C: A] :
% 2.72/1.61 ( ( plus_plus @ A @ ( times_times @ A @ A2 @ E ) @ ( plus_plus @ A @ ( times_times @ A @ B @ E ) @ C ) )
% 2.72/1.61 = ( plus_plus @ A @ ( times_times @ A @ ( plus_plus @ A @ A2 @ B ) @ E ) @ C ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_241_comm__monoid__mult__class_Omult__1,axiom,
% 2.72/1.61 ! [A: $tType] :
% 2.72/1.61 ( ( comm_monoid_mult @ A )
% 2.72/1.61 => ! [A2: A] :
% 2.72/1.61 ( ( times_times @ A @ ( one_one @ A ) @ A2 )
% 2.72/1.61 = A2 ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_242_mult_Ocomm__neutral,axiom,
% 2.72/1.61 ! [A: $tType] :
% 2.72/1.61 ( ( comm_monoid_mult @ A )
% 2.72/1.61 => ! [A2: A] :
% 2.72/1.61 ( ( times_times @ A @ A2 @ ( one_one @ A ) )
% 2.72/1.61 = A2 ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_243_left__diff__distrib,axiom,
% 2.72/1.61 ! [A: $tType] :
% 2.72/1.61 ( ( ring @ A )
% 2.72/1.61 => ! [A2: A,B: A,C: A] :
% 2.72/1.61 ( ( times_times @ A @ ( minus_minus @ A @ A2 @ B ) @ C )
% 2.72/1.61 = ( minus_minus @ A @ ( times_times @ A @ A2 @ C ) @ ( times_times @ A @ B @ C ) ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_244_right__diff__distrib,axiom,
% 2.72/1.61 ! [A: $tType] :
% 2.72/1.61 ( ( ring @ A )
% 2.72/1.61 => ! [A2: A,B: A,C: A] :
% 2.72/1.61 ( ( times_times @ A @ A2 @ ( minus_minus @ A @ B @ C ) )
% 2.72/1.61 = ( minus_minus @ A @ ( times_times @ A @ A2 @ B ) @ ( times_times @ A @ A2 @ C ) ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_245_left__diff__distrib_H,axiom,
% 2.72/1.61 ! [A: $tType] :
% 2.72/1.61 ( ( comm_s1003936772cancel @ A )
% 2.72/1.61 => ! [B: A,C: A,A2: A] :
% 2.72/1.61 ( ( times_times @ A @ ( minus_minus @ A @ B @ C ) @ A2 )
% 2.72/1.61 = ( minus_minus @ A @ ( times_times @ A @ B @ A2 ) @ ( times_times @ A @ C @ A2 ) ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_246_right__diff__distrib_H,axiom,
% 2.72/1.61 ! [A: $tType] :
% 2.72/1.61 ( ( comm_s1003936772cancel @ A )
% 2.72/1.61 => ! [A2: A,B: A,C: A] :
% 2.72/1.61 ( ( times_times @ A @ A2 @ ( minus_minus @ A @ B @ C ) )
% 2.72/1.61 = ( minus_minus @ A @ ( times_times @ A @ A2 @ B ) @ ( times_times @ A @ A2 @ C ) ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_247_abs__mult,axiom,
% 2.72/1.61 ! [A: $tType] :
% 2.72/1.61 ( ( idom_abs_sgn @ A )
% 2.72/1.61 => ! [A2: A,B: A] :
% 2.72/1.61 ( ( abs_abs @ A @ ( times_times @ A @ A2 @ B ) )
% 2.72/1.61 = ( times_times @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B ) ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_248_mult_Oleft__commute,axiom,
% 2.72/1.61 ! [A: $tType] :
% 2.72/1.61 ( ( ab_semigroup_mult @ A )
% 2.72/1.61 => ! [B: A,A2: A,C: A] :
% 2.72/1.61 ( ( times_times @ A @ B @ ( times_times @ A @ A2 @ C ) )
% 2.72/1.61 = ( times_times @ A @ A2 @ ( times_times @ A @ B @ C ) ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_249_mult_Ocommute,axiom,
% 2.72/1.61 ! [A: $tType] :
% 2.72/1.61 ( ( ab_semigroup_mult @ A )
% 2.72/1.61 => ( ( times_times @ A )
% 2.72/1.61 = ( ^ [A3: A,B3: A] : ( times_times @ A @ B3 @ A3 ) ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_250_mult_Oassoc,axiom,
% 2.72/1.61 ! [A: $tType] :
% 2.72/1.61 ( ( semigroup_mult @ A )
% 2.72/1.61 => ! [A2: A,B: A,C: A] :
% 2.72/1.61 ( ( times_times @ A @ ( times_times @ A @ A2 @ B ) @ C )
% 2.72/1.61 = ( times_times @ A @ A2 @ ( times_times @ A @ B @ C ) ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_251_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 2.72/1.61 ! [A: $tType] :
% 2.72/1.61 ( ( ab_semigroup_mult @ A )
% 2.72/1.61 => ! [A2: A,B: A,C: A] :
% 2.72/1.61 ( ( times_times @ A @ ( times_times @ A @ A2 @ B ) @ C )
% 2.72/1.61 = ( times_times @ A @ A2 @ ( times_times @ A @ B @ C ) ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(fact_252_mult__mono,axiom,
% 2.72/1.61 ! [A: $tType] :
% 2.72/1.61 ( ( ordered_semiring @ A )
% 2.72/1.61 => ! [A2: A,B: A,C: A,D: A] :
% 2.72/1.61 ( ( ord_less_eq @ A @ A2 @ B )
% 2.72/1.61 => ( ( ord_less_eq @ A @ C @ D )
% 2.72/1.61 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B )
% 2.72/1.61 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C )
% 2.72/1.61 => ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C ) @ ( times_times @ A @ B @ D ) ) ) ) ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(tcon_fun___Groups_Ominus,axiom,
% 2.72/1.61 ! [A6: $tType,A7: $tType] :
% 2.72/1.61 ( ( minus @ A7 )
% 2.72/1.61 => ( minus @ ( A6 > A7 ) ) ) ).
% 2.72/1.61
% 2.72/1.61 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
% 2.72/1.61 ordere516151231imp_le @ nat ).
% 2.72/1.61
% 2.72/1.61 thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel,axiom,
% 2.72/1.61 semiri1923998003cancel @ nat ).
% 2.72/1.61
% 2.72/1.61 thf(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__diff,axiom,
% 2.72/1.61 ordere623563068d_diff @ nat ).
% 2.72/1.61
% 2.72/1.61 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
% 2.72/1.61 ordere236663937imp_le @ nat ).
% 2.72/1.61
% 2.72/1.61 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 2.72/1.61 canoni770627133id_add @ nat ).
% 2.72/1.61
% 2.72/1.61 thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring,axiom,
% 2.72/1.61 linord1659791738miring @ nat ).
% 2.72/1.61
% 2.72/1.61 thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors,axiom,
% 2.72/1.61 semiri1193490041visors @ nat ).
% 2.72/1.61
% 2.72/1.61 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add,axiom,
% 2.72/1.61 ordere779506340up_add @ nat ).
% 2.72/1.61
% 2.72/1.61 thf(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add,axiom,
% 2.72/1.61 ordere216010020id_add @ nat ).
% 2.72/1.61
% 2.72/1.61 thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add,axiom,
% 2.72/1.61 cancel146912293up_add @ nat ).
% 2.72/1.61
% 2.72/1.61 thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add,axiom,
% 2.72/1.61 cancel1352612707id_add @ nat ).
% 2.72/1.61
% 2.72/1.61 thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1__cancel,axiom,
% 2.72/1.61 comm_s1003936772cancel @ nat ).
% 2.72/1.61
% 2.72/1.61 thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add,axiom,
% 2.72/1.61 cancel_semigroup_add @ nat ).
% 2.72/1.61
% 2.72/1.61 thf(tcon_Nat_Onat___Rings_Olinordered__semidom,axiom,
% 2.72/1.61 linordered_semidom @ nat ).
% 2.72/1.61
% 2.72/1.61 thf(tcon_Nat_Onat___Groups_Oab__semigroup__mult,axiom,
% 2.72/1.61 ab_semigroup_mult @ nat ).
% 2.72/1.61
% 2.72/1.61 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__mult,axiom,
% 2.72/1.61 comm_monoid_mult @ nat ).
% 2.72/1.61
% 2.72/1.61 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__diff,axiom,
% 2.72/1.61 comm_monoid_diff @ nat ).
% 2.72/1.61
% 2.72/1.61 thf(tcon_Nat_Onat___Groups_Oab__semigroup__add,axiom,
% 2.72/1.61 ab_semigroup_add @ nat ).
% 2.72/1.61
% 2.72/1.61 thf(tcon_Nat_Onat___Rings_Oordered__semiring,axiom,
% 2.72/1.61 ordered_semiring @ nat ).
% 2.72/1.61
% 2.72/1.61 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add,axiom,
% 2.72/1.61 comm_monoid_add @ nat ).
% 2.72/1.61
% 2.72/1.61 thf(tcon_Nat_Onat___Groups_Osemigroup__mult,axiom,
% 2.72/1.61 semigroup_mult @ nat ).
% 2.72/1.61
% 2.72/1.61 thf(tcon_Nat_Onat___Groups_Osemigroup__add,axiom,
% 2.72/1.61 semigroup_add @ nat ).
% 2.72/1.61
% 2.72/1.61 thf(tcon_Nat_Onat___Rings_Ocomm__semiring,axiom,
% 2.72/1.61 comm_semiring @ nat ).
% 2.72/1.61
% 2.72/1.61 thf(tcon_Nat_Onat___Rings_Ozero__neq__one,axiom,
% 2.72/1.61 zero_neq_one @ nat ).
% 2.72/1.61
% 2.72/1.61 thf(tcon_Nat_Onat___Groups_Omonoid__mult,axiom,
% 2.72/1.61 monoid_mult @ nat ).
% 2.72/1.61
% 2.72/1.61 thf(tcon_Nat_Onat___Groups_Omonoid__add,axiom,
% 2.72/1.61 monoid_add @ nat ).
% 2.72/1.61
% 2.72/1.61 thf(tcon_Nat_Onat___Rings_Omult__zero,axiom,
% 2.72/1.61 mult_zero @ nat ).
% 2.72/1.61
% 2.72/1.61 thf(tcon_Nat_Onat___Rings_Osemiring,axiom,
% 2.72/1.61 semiring @ nat ).
% 2.72/1.61
% 2.72/1.61 thf(tcon_Nat_Onat___Groups_Ominus_1,axiom,
% 2.72/1.61 minus @ nat ).
% 2.72/1.61
% 2.72/1.61 thf(tcon_Nat_Onat___Groups_Ozero,axiom,
% 2.72/1.61 zero @ nat ).
% 2.72/1.61
% 2.72/1.61 thf(tcon_Nat_Onat___Groups_Oone,axiom,
% 2.72/1.61 one @ nat ).
% 2.72/1.61
% 2.72/1.61 thf(tcon_HOL_Obool___Groups_Ominus_2,axiom,
% 2.72/1.61 minus @ $o ).
% 2.72/1.61
% 2.72/1.61 thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__monoid__add__imp__le_3,axiom,
% 2.72/1.61 ordere516151231imp_le @ real ).
% 2.72/1.61
% 2.72/1.61 thf(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors__cancel_4,axiom,
% 2.72/1.61 semiri1923998003cancel @ real ).
% 2.72/1.61
% 2.72/1.61 thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add__imp__le_5,axiom,
% 2.72/1.61 ordere236663937imp_le @ real ).
% 2.72/1.61
% 2.72/1.61 thf(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_6,axiom,
% 2.72/1.61 linord1659791738miring @ real ).
% 2.72/1.61
% 2.72/1.61 thf(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors_7,axiom,
% 2.72/1.61 semiri1193490041visors @ real ).
% 2.72/1.61
% 2.72/1.61 thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add_8,axiom,
% 2.72/1.61 ordere779506340up_add @ real ).
% 2.72/1.61
% 2.72/1.61 thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add__abs,axiom,
% 2.72/1.61 ordere142940540dd_abs @ real ).
% 2.72/1.61
% 2.72/1.61 thf(tcon_Real_Oreal___Groups_Oordered__comm__monoid__add_9,axiom,
% 2.72/1.61 ordere216010020id_add @ real ).
% 2.72/1.61
% 2.72/1.61 thf(tcon_Real_Oreal___Groups_Olinordered__ab__group__add,axiom,
% 2.72/1.61 linord219039673up_add @ real ).
% 2.72/1.61
% 2.72/1.61 thf(tcon_Real_Oreal___Groups_Ocancel__ab__semigroup__add_10,axiom,
% 2.72/1.61 cancel146912293up_add @ real ).
% 2.72/1.61
% 2.72/1.61 thf(tcon_Real_Oreal___Rings_Oring__1__no__zero__divisors,axiom,
% 2.72/1.61 ring_11004092258visors @ real ).
% 2.72/1.61
% 2.72/1.61 thf(tcon_Real_Oreal___Groups_Ocancel__comm__monoid__add_11,axiom,
% 2.72/1.61 cancel1352612707id_add @ real ).
% 2.72/1.61
% 2.72/1.61 thf(tcon_Real_Oreal___Rings_Ocomm__semiring__1__cancel_12,axiom,
% 2.72/1.61 comm_s1003936772cancel @ real ).
% 2.72/1.61
% 2.72/1.61 thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add,axiom,
% 2.72/1.61 ordered_ab_group_add @ real ).
% 2.72/1.61
% 2.72/1.61 thf(tcon_Real_Oreal___Groups_Ocancel__semigroup__add_13,axiom,
% 2.72/1.61 cancel_semigroup_add @ real ).
% 2.72/1.61
% 2.72/1.61 thf(tcon_Real_Oreal___Rings_Olinordered__semidom_14,axiom,
% 2.72/1.61 linordered_semidom @ real ).
% 2.72/1.61
% 2.72/1.61 thf(tcon_Real_Oreal___Groups_Oab__semigroup__mult_15,axiom,
% 2.72/1.61 ab_semigroup_mult @ real ).
% 2.72/1.61
% 2.72/1.61 thf(tcon_Real_Oreal___Groups_Ocomm__monoid__mult_16,axiom,
% 2.72/1.61 comm_monoid_mult @ real ).
% 2.72/1.61
% 2.72/1.61 thf(tcon_Real_Oreal___Groups_Oab__semigroup__add_17,axiom,
% 2.72/1.61 ab_semigroup_add @ real ).
% 2.72/1.61
% 2.72/1.61 thf(tcon_Real_Oreal___Rings_Oordered__semiring_18,axiom,
% 2.72/1.61 ordered_semiring @ real ).
% 2.72/1.61
% 2.72/1.61 thf(tcon_Real_Oreal___Groups_Ocomm__monoid__add_19,axiom,
% 2.72/1.61 comm_monoid_add @ real ).
% 2.72/1.61
% 2.72/1.61 thf(tcon_Real_Oreal___Rings_Olinordered__idom,axiom,
% 2.72/1.61 linordered_idom @ real ).
% 2.72/1.61
% 2.72/1.61 thf(tcon_Real_Oreal___Groups_Osemigroup__mult_20,axiom,
% 2.72/1.61 semigroup_mult @ real ).
% 2.72/1.61
% 2.72/1.61 thf(tcon_Real_Oreal___Groups_Osemigroup__add_21,axiom,
% 2.72/1.61 semigroup_add @ real ).
% 2.72/1.61
% 2.72/1.61 thf(tcon_Real_Oreal___Rings_Ocomm__semiring_22,axiom,
% 2.72/1.61 comm_semiring @ real ).
% 2.72/1.61
% 2.72/1.61 thf(tcon_Real_Oreal___Groups_Oab__group__add,axiom,
% 2.72/1.61 ab_group_add @ real ).
% 2.72/1.61
% 2.72/1.61 thf(tcon_Real_Oreal___Rings_Ozero__neq__one_23,axiom,
% 2.72/1.61 zero_neq_one @ real ).
% 2.72/1.61
% 2.72/1.61 thf(tcon_Real_Oreal___Rings_Oidom__abs__sgn,axiom,
% 2.72/1.61 idom_abs_sgn @ real ).
% 2.72/1.61
% 2.72/1.61 thf(tcon_Real_Oreal___Groups_Omonoid__mult_24,axiom,
% 2.72/1.61 monoid_mult @ real ).
% 2.72/1.61
% 2.72/1.61 thf(tcon_Real_Oreal___Groups_Omonoid__add_25,axiom,
% 2.72/1.61 monoid_add @ real ).
% 2.72/1.61
% 2.72/1.61 thf(tcon_Real_Oreal___Groups_Ogroup__add,axiom,
% 2.72/1.61 group_add @ real ).
% 2.72/1.61
% 2.72/1.61 thf(tcon_Real_Oreal___Rings_Omult__zero_26,axiom,
% 2.72/1.61 mult_zero @ real ).
% 2.72/1.61
% 2.72/1.61 thf(tcon_Real_Oreal___Num_Oneg__numeral,axiom,
% 2.72/1.61 neg_numeral @ real ).
% 2.72/1.61
% 2.72/1.61 thf(tcon_Real_Oreal___Rings_Osemiring_27,axiom,
% 2.72/1.61 semiring @ real ).
% 2.72/1.61
% 2.72/1.61 thf(tcon_Real_Oreal___Groups_Ominus_28,axiom,
% 2.72/1.61 minus @ real ).
% 2.72/1.61
% 2.72/1.61 thf(tcon_Real_Oreal___Groups_Ozero_29,axiom,
% 2.72/1.61 zero @ real ).
% 2.72/1.61
% 2.72/1.61 thf(tcon_Real_Oreal___Rings_Oring,axiom,
% 2.72/1.61 ring @ real ).
% 2.72/1.61
% 2.72/1.61 thf(tcon_Real_Oreal___Groups_Oone_30,axiom,
% 2.72/1.61 one @ real ).
% 2.72/1.61
% 2.72/1.61 thf(conj_0,conjecture,
% 2.72/1.61 ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( genClo1163638703lle_te @ p @ ( plus_plus @ nat @ i @ ( one_one @ nat ) ) ) @ ( genClo1163638703lle_te @ q @ ( plus_plus @ nat @ i @ ( one_one @ nat ) ) ) ) ) @ genClo1278781456e_beta ).
% 2.72/1.61 ------------------------
% 2.72/1.63 % (9209)lrs+10_1:1_c=on:cnfonf=conj_eager:fd=off:fe=off:kws=frequency:spb=intro:i=4:si=on:rtra=on_0 on DTF2THF_9062 for (2999ds/4Mi)
% 2.72/1.63 % (9210)dis+1010_1:1_au=on:cbe=off:chr=on:fsr=off:hfsq=on:nm=64:sos=theory:sp=weighted_frequency:i=27:si=on:rtra=on_0 on DTF2THF_9062 for (2999ds/27Mi)
% 2.72/1.63 % (9211)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_9062 for (2999ds/2Mi)
% 2.72/1.63 % (9214)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_9062 for (2999ds/18Mi)
% 2.72/1.63 % (9208)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_9062 for (2999ds/183Mi)
% 2.72/1.63 % (9209)Instruction limit reached!
% 2.72/1.63 % (9209)------------------------------
% 2.72/1.63 % (9209)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.63 % (9209)Termination reason: Unknown
% 2.72/1.63 % (9209)Termination phase: shuffling
% 2.72/1.63
% 2.72/1.63 % (9209)Memory used [KB]: 1407
% 2.72/1.63 % (9209)Time elapsed: 0.003 s
% 2.72/1.63 % (9209)Instructions burned: 6 (million)
% 2.72/1.63 % (9209)------------------------------
% 2.72/1.63 % (9209)------------------------------
% 2.72/1.63 % (9211)Instruction limit reached!
% 2.72/1.63 % (9211)------------------------------
% 2.72/1.63 % (9211)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.63 % (9211)Termination reason: Unknown
% 2.72/1.63 % (9211)Termination phase: shuffling
% 2.72/1.63
% 2.72/1.63 % (9211)Memory used [KB]: 1279
% 2.72/1.63 % (9211)Time elapsed: 0.002 s
% 2.72/1.63 % (9211)Instructions burned: 3 (million)
% 2.72/1.63 % (9211)------------------------------
% 2.72/1.63 % (9211)------------------------------
% 2.72/1.63 % (9214)Instruction limit reached!
% 2.72/1.63 % (9214)------------------------------
% 2.72/1.63 % (9214)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.63 % (9214)Termination reason: Unknown
% 2.72/1.63 % (9214)Termination phase: Property scanning
% 2.72/1.63
% 2.72/1.63 % (9214)Memory used [KB]: 1663
% 2.72/1.63 % (9214)Time elapsed: 0.008 s
% 2.72/1.63 % (9214)Instructions burned: 21 (million)
% 2.72/1.63 % (9212)lrs+1002_1:128_aac=none:au=on:cnfonf=lazy_not_gen_be_off:sos=all:i=2:si=on:rtra=on_0 on DTF2THF_9062 for (2999ds/2Mi)
% 2.72/1.63 % (9214)------------------------------
% 2.72/1.63 % (9214)------------------------------
% 2.72/1.63 % (9213)lrs+1002_1:1_au=on:bd=off:e2e=on:sd=2:sos=on:ss=axioms:i=275:si=on:rtra=on_0 on DTF2THF_9062 for (2999ds/275Mi)
% 2.72/1.63 % (9212)Instruction limit reached!
% 2.72/1.63 % (9212)------------------------------
% 2.72/1.63 % (9212)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.63 % (9212)Termination reason: Unknown
% 2.72/1.63 % (9212)Termination phase: shuffling
% 2.72/1.63
% 2.72/1.63 % (9212)Memory used [KB]: 1279
% 2.72/1.63 % (9212)Time elapsed: 0.003 s
% 2.72/1.63 % (9212)Instructions burned: 2 (million)
% 2.72/1.63 % (9212)------------------------------
% 2.72/1.63 % (9212)------------------------------
% 2.72/1.64 % (9210)Instruction limit reached!
% 2.72/1.64 % (9210)------------------------------
% 2.72/1.64 % (9210)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.64 % (9210)Termination reason: Unknown
% 2.72/1.64 % (9210)Termination phase: Property scanning
% 2.72/1.64
% 2.72/1.64 % (9210)Memory used [KB]: 1663
% 2.72/1.64 % (9210)Time elapsed: 0.010 s
% 2.72/1.64 % (9210)Instructions burned: 28 (million)
% 2.72/1.64 % (9210)------------------------------
% 2.72/1.64 % (9210)------------------------------
% 2.72/1.64 % (9215)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_9062 for (2999ds/3Mi)
% 2.72/1.64 % (9216)lrs+1002_1:1_cnfonf=lazy_not_be_gen:hud=14:prag=on:sp=weighted_frequency:tnu=1:i=37:si=on:rtra=on_0 on DTF2THF_9062 for (2999ds/37Mi)
% 2.72/1.64 % (9215)Instruction limit reached!
% 2.72/1.64 % (9215)------------------------------
% 2.72/1.64 % (9215)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.64 % (9215)Termination reason: Unknown
% 2.72/1.64 % (9215)Termination phase: shuffling
% 2.72/1.64
% 2.72/1.64 % (9215)Memory used [KB]: 1407
% 2.72/1.64 % (9215)Time elapsed: 0.002 s
% 2.72/1.64 % (9215)Instructions burned: 4 (million)
% 2.72/1.64 % (9215)------------------------------
% 2.72/1.64 % (9215)------------------------------
% 2.72/1.64 % (9217)lrs+2_16:1_acc=model:au=on:bd=off:c=on:e2e=on:nm=2:sos=all:i=15:si=on:rtra=on_0 on DTF2THF_9062 for (2999ds/15Mi)
% 2.72/1.65 % (9219)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=1041:si=on:rtra=on_0 on DTF2THF_9062 for (2999ds/1041Mi)
% 2.72/1.65 % (9220)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_9062 for (2999ds/7Mi)
% 2.72/1.65 % (9216)Instruction limit reached!
% 2.72/1.65 % (9216)------------------------------
% 2.72/1.65 % (9216)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.65 % (9216)Termination reason: Unknown
% 2.72/1.65 % (9216)Termination phase: Property scanning
% 2.72/1.65
% 2.72/1.65 % (9216)Memory used [KB]: 2046
% 2.72/1.65 % (9216)Time elapsed: 0.013 s
% 2.72/1.65 % (9216)Instructions burned: 39 (million)
% 2.72/1.65 % (9216)------------------------------
% 2.72/1.65 % (9216)------------------------------
% 2.72/1.65 % (9217)Instruction limit reached!
% 2.72/1.65 % (9217)------------------------------
% 2.72/1.65 % (9217)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.65 % (9217)Termination reason: Unknown
% 2.72/1.65 % (9217)Termination phase: shuffling
% 2.72/1.65
% 2.72/1.65 % (9217)Memory used [KB]: 1663
% 2.72/1.65 % (9217)Time elapsed: 0.007 s
% 2.72/1.65 % (9217)Instructions burned: 18 (million)
% 2.72/1.65 % (9217)------------------------------
% 2.72/1.65 % (9217)------------------------------
% 2.72/1.65 % (9220)Instruction limit reached!
% 2.72/1.65 % (9220)------------------------------
% 2.72/1.65 % (9220)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.65 % (9220)Termination reason: Unknown
% 2.72/1.65 % (9220)Termination phase: shuffling
% 2.72/1.65
% 2.72/1.65 % (9220)Memory used [KB]: 1407
% 2.72/1.65 % (9220)Time elapsed: 0.004 s
% 2.72/1.65 % (9220)Instructions burned: 9 (million)
% 2.72/1.65 % (9220)------------------------------
% 2.72/1.65 % (9220)------------------------------
% 2.72/1.65 % (9218)dis+21_1:1_cbe=off:cnfonf=off:fs=off:fsr=off:hud=1:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_9062 for (2999ds/3Mi)
% 2.72/1.65 % (9218)Instruction limit reached!
% 2.72/1.65 % (9218)------------------------------
% 2.72/1.65 % (9218)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.65 % (9218)Termination reason: Unknown
% 2.72/1.65 % (9218)Termination phase: shuffling
% 2.72/1.65
% 2.72/1.65 % (9218)Memory used [KB]: 1407
% 2.72/1.65 % (9218)Time elapsed: 0.004 s
% 2.72/1.65 % (9218)Instructions burned: 4 (million)
% 2.72/1.65 % (9218)------------------------------
% 2.72/1.65 % (9218)------------------------------
% 2.72/1.66 % (9222)lrs+21_1:1_au=on:cnfonf=off:fd=preordered:fe=off:fsr=off:hud=11:inj=on:kws=precedence:s2pl=no:sp=weighted_frequency:tgt=full:i=3:si=on:rtra=on_0 on DTF2THF_9062 for (2999ds/3Mi)
% 2.72/1.66 % (9221)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_9062 for (2999ds/16Mi)
% 2.72/1.66 % (9223)lrs+2_1:1_apa=on:au=on:bd=preordered:cnfonf=off:cs=on:ixr=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_9062 for (2999ds/3Mi)
% 2.72/1.66 % (9222)Instruction limit reached!
% 2.72/1.66 % (9222)------------------------------
% 2.72/1.66 % (9222)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.66 % (9222)Termination reason: Unknown
% 2.72/1.66 % (9222)Termination phase: shuffling
% 2.72/1.66
% 2.72/1.66 % (9222)Memory used [KB]: 1407
% 2.72/1.66 % (9222)Time elapsed: 0.002 s
% 2.72/1.66 % (9222)Instructions burned: 4 (million)
% 2.72/1.66 % (9222)------------------------------
% 2.72/1.66 % (9222)------------------------------
% 2.72/1.66 % (9223)Instruction limit reached!
% 2.72/1.66 % (9223)------------------------------
% 2.72/1.66 % (9223)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.66 % (9223)Termination reason: Unknown
% 2.72/1.66 % (9223)Termination phase: shuffling
% 2.72/1.66
% 2.72/1.66 % (9223)Memory used [KB]: 1407
% 2.72/1.66 % (9223)Time elapsed: 0.002 s
% 2.72/1.66 % (9223)Instructions burned: 3 (million)
% 2.72/1.66 % (9223)------------------------------
% 2.72/1.66 % (9223)------------------------------
% 2.72/1.66 % (9213)First to succeed.
% 2.72/1.66 % (9213)Refutation found. Thanks to Tanya!
% 2.72/1.66 % SZS status Theorem for DTF2THF_9062
% 2.72/1.66 % SZS output start Proof for DTF2THF_9062
% 2.72/1.66 thf(type_def_5, type, nat: $tType).
% 2.72/1.66 thf(type_def_6, type, real: $tType).
% 2.72/1.66 thf(func_def_0, type, real: $tType).
% 2.72/1.66 thf(func_def_1, type, nat: $tType).
% 2.72/1.66 thf(func_def_2, type, type: !>[X0: $tType]:($o)).
% 2.72/1.66 thf(func_def_3, type, one: !>[X0: $tType]:($o)).
% 2.72/1.66 thf(func_def_4, type, ring: !>[X0: $tType]:($o)).
% 2.72/1.66 thf(func_def_5, type, zero: !>[X0: $tType]:($o)).
% 2.72/1.66 thf(func_def_6, type, minus: !>[X0: $tType]:($o)).
% 2.72/1.66 thf(func_def_7, type, semiring: !>[X0: $tType]:($o)).
% 2.72/1.66 thf(func_def_8, type, neg_numeral: !>[X0: $tType]:($o)).
% 2.72/1.66 thf(func_def_9, type, mult_zero: !>[X0: $tType]:($o)).
% 2.72/1.66 thf(func_def_10, type, group_add: !>[X0: $tType]:($o)).
% 2.72/1.66 thf(func_def_11, type, monoid_add: !>[X0: $tType]:($o)).
% 2.72/1.66 thf(func_def_12, type, monoid_mult: !>[X0: $tType]:($o)).
% 2.72/1.66 thf(func_def_13, type, idom_abs_sgn: !>[X0: $tType]:($o)).
% 2.72/1.66 thf(func_def_14, type, zero_neq_one: !>[X0: $tType]:($o)).
% 2.72/1.66 thf(func_def_15, type, ab_group_add: !>[X0: $tType]:($o)).
% 2.72/1.66 thf(func_def_16, type, comm_semiring: !>[X0: $tType]:($o)).
% 2.72/1.66 thf(func_def_17, type, semigroup_add: !>[X0: $tType]:($o)).
% 2.72/1.66 thf(func_def_18, type, semigroup_mult: !>[X0: $tType]:($o)).
% 2.72/1.66 thf(func_def_19, type, linordered_idom: !>[X0: $tType]:($o)).
% 2.72/1.66 thf(func_def_20, type, comm_monoid_add: !>[X0: $tType]:($o)).
% 2.72/1.66 thf(func_def_21, type, ordered_semiring: !>[X0: $tType]:($o)).
% 2.72/1.66 thf(func_def_22, type, ab_semigroup_add: !>[X0: $tType]:($o)).
% 2.72/1.66 thf(func_def_23, type, comm_monoid_diff: !>[X0: $tType]:($o)).
% 2.72/1.66 thf(func_def_24, type, comm_monoid_mult: !>[X0: $tType]:($o)).
% 2.72/1.66 thf(func_def_25, type, ab_semigroup_mult: !>[X0: $tType]:($o)).
% 2.72/1.66 thf(func_def_26, type, linordered_semidom: !>[X0: $tType]:($o)).
% 2.72/1.66 thf(func_def_27, type, cancel_semigroup_add: !>[X0: $tType]:($o)).
% 2.72/1.66 thf(func_def_28, type, ordered_ab_group_add: !>[X0: $tType]:($o)).
% 2.72/1.66 thf(func_def_29, type, comm_s1003936772cancel: !>[X0: $tType]:($o)).
% 2.72/1.66 thf(func_def_30, type, cancel1352612707id_add: !>[X0: $tType]:($o)).
% 2.72/1.66 thf(func_def_31, type, ring_11004092258visors: !>[X0: $tType]:($o)).
% 2.72/1.66 thf(func_def_32, type, cancel146912293up_add: !>[X0: $tType]:($o)).
% 2.72/1.66 thf(func_def_33, type, linord219039673up_add: !>[X0: $tType]:($o)).
% 2.72/1.66 thf(func_def_34, type, ordere216010020id_add: !>[X0: $tType]:($o)).
% 2.72/1.66 thf(func_def_35, type, ordere142940540dd_abs: !>[X0: $tType]:($o)).
% 2.72/1.66 thf(func_def_36, type, ordere779506340up_add: !>[X0: $tType]:($o)).
% 2.72/1.66 thf(func_def_37, type, semiri1193490041visors: !>[X0: $tType]:($o)).
% 2.72/1.66 thf(func_def_38, type, linord1659791738miring: !>[X0: $tType]:($o)).
% 2.72/1.66 thf(func_def_39, type, canoni770627133id_add: !>[X0: $tType]:($o)).
% 2.72/1.66 thf(func_def_40, type, ordere236663937imp_le: !>[X0: $tType]:($o)).
% 2.72/1.66 thf(func_def_41, type, ordere623563068d_diff: !>[X0: $tType]:($o)).
% 2.72/1.66 thf(func_def_42, type, semiri1923998003cancel: !>[X0: $tType]:($o)).
% 2.72/1.66 thf(func_def_43, type, ordere516151231imp_le: !>[X0: $tType]:($o)).
% 2.72/1.66 thf(func_def_44, type, genClo1161277105lle_PC: nat > real > real).
% 2.72/1.66 thf(func_def_45, type, genClo1278781456e_beta: real).
% 2.72/1.66 thf(func_def_46, type, genClo872980712lle_mu: real).
% 2.72/1.66 thf(func_def_47, type, genClo1144207539le_rho: real).
% 2.72/1.66 thf(func_def_48, type, genClo1015804716orrect: nat > real > $o).
% 2.72/1.66 thf(func_def_49, type, genClo293725282kRead2: (nat > real) > (nat > real) > real > (nat > $o) > $o).
% 2.72/1.66 thf(func_def_50, type, genClo1650508560e_rmax: real).
% 2.72/1.66 thf(func_def_51, type, genClo1651033342e_rmin: real).
% 2.72/1.66 thf(func_def_52, type, genClo1163638703lle_te: nat > nat > real).
% 2.72/1.66 thf(func_def_53, type, abs_abs: !>[X0: $tType]:(X0 > X0)).
% 2.72/1.66 thf(func_def_54, type, minus_minus: !>[X0: $tType]:(X0 > X0 > X0)).
% 2.72/1.66 thf(func_def_55, type, one_one: !>[X0: $tType]:(X0)).
% 2.72/1.66 thf(func_def_56, type, plus_plus: !>[X0: $tType]:(X0 > X0 > X0)).
% 2.72/1.66 thf(func_def_57, type, times_times: !>[X0: $tType]:(X0 > X0 > X0)).
% 2.72/1.66 thf(func_def_58, type, zero_zero: !>[X0: $tType]:(X0)).
% 2.72/1.66 thf(func_def_59, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.72/1.66 thf(func_def_60, type, i: nat).
% 2.72/1.66 thf(func_def_61, type, l: nat).
% 2.72/1.66 thf(func_def_62, type, p: nat).
% 2.72/1.66 thf(func_def_63, type, q: nat).
% 2.72/1.66 thf(func_def_77, type, sK0: nat > (nat > $o) > nat).
% 2.72/1.66 thf(func_def_78, type, sK1: (nat > $o) > nat).
% 2.72/1.66 thf(func_def_79, type, sK2: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 2.72/1.66 thf(func_def_80, type, sK3: nat > nat > nat).
% 2.72/1.66 thf(func_def_82, type, ph5: !>[X0: $tType]:(X0)).
% 2.72/1.66 thf(f1126,plain,(
% 2.72/1.66 $false),
% 2.72/1.66 inference(subsumption_resolution,[status(thm)],[f1125,f1105])).
% 2.72/1.66 thf(f1105,plain,(
% 2.72/1.66 ((genClo1015804716orrect @ p @ (genClo1163638703lle_te @ p @ (plus_plus @ nat @ i @ one_one @ nat))) = $true)),
% 2.72/1.66 inference(cnf_transformation,[status(thm)],[f473])).
% 2.72/1.66 thf(f473,plain,(
% 2.72/1.66 ((genClo1015804716orrect @ p @ (genClo1163638703lle_te @ p @ (plus_plus @ nat @ i @ one_one @ nat))) = $true)),
% 2.72/1.66 inference(fool_elimination,[status(thm)],[f472])).
% 2.72/1.66 thf(f472,plain,(
% 2.72/1.66 (genClo1015804716orrect @ p @ (genClo1163638703lle_te @ p @ (plus_plus @ nat @ i @ one_one @ nat)))),
% 2.72/1.66 inference(rectify,[status(thm)],[f2])).
% 2.72/1.66 thf(f2,axiom,(
% 2.72/1.66 (genClo1015804716orrect @ p @ (genClo1163638703lle_te @ p @ (plus_plus @ nat @ i @ one_one @ nat)))),
% 2.72/1.66 file('/export/starexec/sandbox/tmp/tmp.XXNzzqTI4f/DTF2THF_9062.p',fact_1_corr__p)).
% 2.72/1.66 thf(f1125,plain,(
% 2.72/1.66 ((genClo1015804716orrect @ p @ (genClo1163638703lle_te @ p @ (plus_plus @ nat @ i @ one_one @ nat))) != $true)),
% 2.72/1.66 inference(subsumption_resolution,[status(thm)],[f1124,f1098])).
% 2.72/1.66 thf(f1098,plain,(
% 2.72/1.66 ((genClo1015804716orrect @ q @ (genClo1163638703lle_te @ q @ (plus_plus @ nat @ i @ one_one @ nat))) = $true)),
% 2.72/1.66 inference(cnf_transformation,[status(thm)],[f629])).
% 2.72/1.66 thf(f629,plain,(
% 2.72/1.66 ((genClo1015804716orrect @ q @ (genClo1163638703lle_te @ q @ (plus_plus @ nat @ i @ one_one @ nat))) = $true)),
% 2.72/1.66 inference(fool_elimination,[status(thm)],[f628])).
% 2.72/1.66 thf(f628,plain,(
% 2.72/1.66 (genClo1015804716orrect @ q @ (genClo1163638703lle_te @ q @ (plus_plus @ nat @ i @ one_one @ nat)))),
% 2.72/1.66 inference(rectify,[status(thm)],[f3])).
% 2.72/1.66 thf(f3,axiom,(
% 2.72/1.66 (genClo1015804716orrect @ q @ (genClo1163638703lle_te @ q @ (plus_plus @ nat @ i @ one_one @ nat)))),
% 2.72/1.66 file('/export/starexec/sandbox/tmp/tmp.XXNzzqTI4f/DTF2THF_9062.p',fact_2_corr__q__tq)).
% 2.72/1.66 thf(f1124,plain,(
% 2.72/1.66 ((genClo1015804716orrect @ q @ (genClo1163638703lle_te @ q @ (plus_plus @ nat @ i @ one_one @ nat))) != $true) | ((genClo1015804716orrect @ p @ (genClo1163638703lle_te @ p @ (plus_plus @ nat @ i @ one_one @ nat))) != $true)),
% 2.72/1.66 inference(trivial_inequality_removal,[status(thm)],[f1123])).
% 2.72/1.66 thf(f1123,plain,(
% 2.72/1.66 ($true != $true) | ((genClo1015804716orrect @ p @ (genClo1163638703lle_te @ p @ (plus_plus @ nat @ i @ one_one @ nat))) != $true) | ((genClo1015804716orrect @ q @ (genClo1163638703lle_te @ q @ (plus_plus @ nat @ i @ one_one @ nat))) != $true)),
% 2.72/1.66 inference(superposition,[status(thm)],[f1085,f1091])).
% 2.72/1.66 thf(f1091,plain,(
% 2.72/1.66 ( ! [X2 : nat,X0 : nat,X1 : nat] : (($true = (ord_less_eq @ real @ (abs_abs @ real @ (minus_minus @ real @ (genClo1163638703lle_te @ X0 @ X2) @ (genClo1163638703lle_te @ X1 @ X2))) @ genClo1278781456e_beta)) | ((genClo1015804716orrect @ X1 @ (genClo1163638703lle_te @ X1 @ X2)) != $true) | ($true != (genClo1015804716orrect @ X0 @ (genClo1163638703lle_te @ X0 @ X2)))) )),
% 2.72/1.66 inference(cnf_transformation,[status(thm)],[f998])).
% 2.72/1.66 thf(f998,plain,(
% 2.72/1.66 ! [X0 : nat,X1 : nat,X2 : nat] : (($true != (genClo1015804716orrect @ X0 @ (genClo1163638703lle_te @ X0 @ X2))) | ($true = (ord_less_eq @ real @ (abs_abs @ real @ (minus_minus @ real @ (genClo1163638703lle_te @ X0 @ X2) @ (genClo1163638703lle_te @ X1 @ X2))) @ genClo1278781456e_beta)) | ((genClo1015804716orrect @ X1 @ (genClo1163638703lle_te @ X1 @ X2)) != $true))),
% 2.72/1.66 inference(flattening,[status(thm)],[f997])).
% 2.72/1.66 thf(f997,plain,(
% 2.72/1.66 ! [X0 : nat,X1 : nat,X2 : nat] : (($true = (ord_less_eq @ real @ (abs_abs @ real @ (minus_minus @ real @ (genClo1163638703lle_te @ X0 @ X2) @ (genClo1163638703lle_te @ X1 @ X2))) @ genClo1278781456e_beta)) | (($true != (genClo1015804716orrect @ X0 @ (genClo1163638703lle_te @ X0 @ X2))) | ((genClo1015804716orrect @ X1 @ (genClo1163638703lle_te @ X1 @ X2)) != $true)))),
% 2.72/1.66 inference(ennf_transformation,[status(thm)],[f549])).
% 2.72/1.66 thf(f549,plain,(
% 2.72/1.66 ! [X0 : nat,X1 : nat,X2 : nat] : ((($true = (genClo1015804716orrect @ X0 @ (genClo1163638703lle_te @ X0 @ X2))) & ((genClo1015804716orrect @ X1 @ (genClo1163638703lle_te @ X1 @ X2)) = $true)) => ($true = (ord_less_eq @ real @ (abs_abs @ real @ (minus_minus @ real @ (genClo1163638703lle_te @ X0 @ X2) @ (genClo1163638703lle_te @ X1 @ X2))) @ genClo1278781456e_beta)))),
% 2.72/1.66 inference(fool_elimination,[status(thm)],[f548])).
% 2.72/1.66 thf(f548,plain,(
% 2.72/1.66 ! [X0 : nat,X1 : nat,X2 : nat] : (((genClo1015804716orrect @ X0 @ (genClo1163638703lle_te @ X0 @ X2)) & (genClo1015804716orrect @ X1 @ (genClo1163638703lle_te @ X1 @ X2))) => (ord_less_eq @ real @ (abs_abs @ real @ (minus_minus @ real @ (genClo1163638703lle_te @ X0 @ X2) @ (genClo1163638703lle_te @ X1 @ X2))) @ genClo1278781456e_beta))),
% 2.72/1.66 inference(rectify,[status(thm)],[f1])).
% 2.72/1.66 thf(f1,axiom,(
% 2.72/1.66 ! [X1 : nat,X2 : nat,X3 : nat] : (((genClo1015804716orrect @ X1 @ (genClo1163638703lle_te @ X1 @ X3)) & (genClo1015804716orrect @ X2 @ (genClo1163638703lle_te @ X2 @ X3))) => (ord_less_eq @ real @ (abs_abs @ real @ (minus_minus @ real @ (genClo1163638703lle_te @ X1 @ X3) @ (genClo1163638703lle_te @ X2 @ X3))) @ genClo1278781456e_beta))),
% 2.72/1.66 file('/export/starexec/sandbox/tmp/tmp.XXNzzqTI4f/DTF2THF_9062.p',fact_0_rts2b)).
% 2.72/1.66 thf(f1085,plain,(
% 2.72/1.66 ((ord_less_eq @ real @ (abs_abs @ real @ (minus_minus @ real @ (genClo1163638703lle_te @ p @ (plus_plus @ nat @ i @ one_one @ nat)) @ (genClo1163638703lle_te @ q @ (plus_plus @ nat @ i @ one_one @ nat)))) @ genClo1278781456e_beta) != $true)),
% 2.72/1.66 inference(cnf_transformation,[status(thm)],[f942])).
% 2.72/1.66 thf(f942,plain,(
% 2.72/1.66 ((ord_less_eq @ real @ (abs_abs @ real @ (minus_minus @ real @ (genClo1163638703lle_te @ p @ (plus_plus @ nat @ i @ one_one @ nat)) @ (genClo1163638703lle_te @ q @ (plus_plus @ nat @ i @ one_one @ nat)))) @ genClo1278781456e_beta) != $true)),
% 2.72/1.66 inference(flattening,[status(thm)],[f403])).
% 2.72/1.66 thf(f403,plain,(
% 2.72/1.66 ~((ord_less_eq @ real @ (abs_abs @ real @ (minus_minus @ real @ (genClo1163638703lle_te @ p @ (plus_plus @ nat @ i @ one_one @ nat)) @ (genClo1163638703lle_te @ q @ (plus_plus @ nat @ i @ one_one @ nat)))) @ genClo1278781456e_beta) = $true)),
% 2.72/1.66 inference(fool_elimination,[status(thm)],[f402])).
% 2.72/1.66 thf(f402,plain,(
% 2.72/1.66 ~(ord_less_eq @ real @ (abs_abs @ real @ (minus_minus @ real @ (genClo1163638703lle_te @ p @ (plus_plus @ nat @ i @ one_one @ nat)) @ (genClo1163638703lle_te @ q @ (plus_plus @ nat @ i @ one_one @ nat)))) @ genClo1278781456e_beta)),
% 2.72/1.66 inference(rectify,[status(thm)],[f326])).
% 2.72/1.66 thf(f326,negated_conjecture,(
% 2.72/1.66 ~(ord_less_eq @ real @ (abs_abs @ real @ (minus_minus @ real @ (genClo1163638703lle_te @ p @ (plus_plus @ nat @ i @ one_one @ nat)) @ (genClo1163638703lle_te @ q @ (plus_plus @ nat @ i @ one_one @ nat)))) @ genClo1278781456e_beta)),
% 2.72/1.66 inference(negated_conjecture,[status(cth)],[f325])).
% 2.72/1.66 thf(f325,conjecture,(
% 2.72/1.66 (ord_less_eq @ real @ (abs_abs @ real @ (minus_minus @ real @ (genClo1163638703lle_te @ p @ (plus_plus @ nat @ i @ one_one @ nat)) @ (genClo1163638703lle_te @ q @ (plus_plus @ nat @ i @ one_one @ nat)))) @ genClo1278781456e_beta)),
% 2.72/1.66 file('/export/starexec/sandbox/tmp/tmp.XXNzzqTI4f/DTF2THF_9062.p',conj_0)).
% 2.72/1.66 % SZS output end Proof for DTF2THF_9062
% 2.72/1.66 % (9213)------------------------------
% 2.72/1.66 % (9213)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.66 % (9213)Termination reason: Refutation
% 2.72/1.66
% 2.72/1.66 % (9213)Memory used [KB]: 6396
% 2.72/1.66 % (9213)Time elapsed: 0.030 s
% 2.72/1.66 % (9213)Instructions burned: 47 (million)
% 2.72/1.66 % (9213)------------------------------
% 2.72/1.66 % (9213)------------------------------
% 2.72/1.66 % (9207)Success in time 0.043 s
%------------------------------------------------------------------------------